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Percolation models

The most common percolation model is to take a regular lattice, like a square lattice, and make it into a random network by randomly "occupying" sites (vertices) or bonds (edges) with a statistically independent probability p. At a critical threshold pc, large clusters and long-range connectivity first appear, and this is called the percolation threshold. Depending on the method for obtaining the random network, one distinguishes between the site percolation threshold and the bond percolation threshold. More general systems have several probabilities p1, p2, etc., and the transition is characterized by a critical surface or manifold. One can also consider continuum systems, such as overlapping disks and spheres placed randomly, or the negative space (Swiss-cheese models).

To understand the threshold, you can consider a quantity such as the probability that there is a continuous path from one boundary to another along occupied sites or bonds—that is, within a single cluster. For example, one can consider a square system, and ask for the probability P that there is a path from the top boundary to the bottom boundary. As a function of the occupation probability p, one finds a sigmoidal plot that goes from P=0 at p=0 to P=1 at p=1. The larger the square is compared to the lattice spacing, the sharper the transition will be. When the system size goes to infinity, P(p) will be a step function at the threshold value pc. For finite large systems, P(pc) is a constant whose value depends upon the shape of the system; for the square system discussed above, P(pc)=1⁄2 exactly for any lattice by a simple symmetry argument.

There are other signatures of the critical threshold. For example, the size distribution (number of clusters of size s) drops off as a power-law for large s at the threshold, ns(pc) ~ s−τ, where τ is a dimension-dependent percolation critical exponents. For an infinite system, the critical threshold corresponds to the first point (as p increases) where the size of the clusters become infinite.

In the systems described so far, it has been assumed that the occupation of a site or bond is completely random—this is the so-called Bernoulli percolation. For a continuum system, random occupancy corresponds to the points being placed by a Poisson process. Further variations involve correlated percolation, such as percolation clusters related to Ising and Potts models of ferromagnets, in which the bonds are put down by the Fortuin–Kasteleyn method.² In bootstrap or k-sat percolation, sites and/or bonds are first occupied and then successively culled from a system if a site does not have at least k neighbors. Another important model of percolation, in a different universality class altogether, is directed percolation, where connectivity along a bond depends upon the direction of the flow. Another variation of recent interest is Explosive Percolation, whose thresholds are listed on that page.

Over the last several decades, a tremendous amount of work has gone into finding exact and approximate values of the percolation thresholds for a variety of these systems. Exact thresholds are only known for certain two-dimensional lattices that can be broken up into a self-dual array, such that under a triangle-triangle transformation, the system remains the same. Studies using numerical methods have led to numerous improvements in algorithms and several theoretical discoveries.

Simple duality in two dimensions implies that all fully triangulated lattices (e.g., the triangular, union jack, cross dual, martini dual and asanoha or 3-12 dual, and the Delaunay triangulation) all have site thresholds of 1⁄2, and self-dual lattices (square, martini-B) have bond thresholds of 1⁄2.

The notation such as (4,82) comes from Grünbaum and Shephard,³ and indicates that around a given vertex, going in the clockwise direction, one encounters first a square and then two octagons. Besides the eleven Archimedean lattices composed of regular polygons with every site equivalent, many other more complicated lattices with sites of different classes have been studied.

Error bars in the last digit or digits are shown by numbers in parentheses. Thus, 0.729724(3) signifies 0.729724 ± 0.000003, and 0.74042195(80) signifies 0.74042195 ± 0.00000080. The error bars variously represent one or two standard deviations in net error (including statistical and expected systematic error), or an empirical confidence interval, depending upon the source.

Percolation on networks

For a random tree-like network (i.e., a connected network with no cycle) without degree-degree correlation, it can be shown that such network can have a giant component, and the percolation threshold (transmission probability) is given by

p c = 1 g 1 ′ ( 1 ) = ⟨ k ⟩ ⟨ k 2 ⟩ − ⟨ k ⟩ {\displaystyle p_{c}={\frac {1}{g_{1}'(1)}}={\frac {\langle k\rangle }{\langle k^{2}\rangle -\langle k\rangle }}} .

Where g 1 ( z ) {\displaystyle g_{1}(z)} is the generating function corresponding to the excess degree distribution, ⟨ k ⟩ {\displaystyle {\langle k\rangle }} is the average degree of the network and ⟨ k 2 ⟩ {\displaystyle {\langle k^{2}\rangle }} is the second moment of the degree distribution. So, for example, for an ER network, since the degree distribution is a Poisson distribution, where ⟨ k 2 ⟩ = ⟨ k ⟩ 2 + ⟨ k ⟩ , {\displaystyle {\langle k^{2}\rangle =\langle k\rangle ^{2}+\langle k\rangle },} the threshold is at p c = ⟨ k ⟩ − 1 {\displaystyle p_{c}={\langle k\rangle }^{-1}} .

In networks with low clustering, 0 < C ≪ 1 {\displaystyle 0<C\ll 1} , the critical point gets scaled by ( 1 − C ) − 1 {\displaystyle (1-C)^{-1}} such that:⁴

p c = 1 1 − C 1 g 1 ′ ( 1 ) . {\displaystyle p_{c}={\frac {1}{1-C}}{\frac {1}{g_{1}'(1)}}.}

This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold, mainly because for a fixed number of links, the clustering structure reinforces the core of the network with the price of diluting the global connections. For networks with high clustering, strong clustering could induce the core–periphery structure, in which the core and periphery might percolate at different critical points, and the above approximate treatment is not applicable.⁵

Percolation in 2D

Thresholds on Archimedean lattices

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
3-12 or super-kagome, (3, 122 )	3	3	0.807900764... = (1 − 2 sin (π/18))1⁄26	0.74042195(80),7 0.74042077(2),8 0.740420800(2),9 0.7404207988509(8),1011 0.740420798850811610(2),12
cross, truncated trihexagonal (4, 6, 12)	3	3	0.746,13 0.750,14 0.747806(4),15 0.7478008(2)16	0.6937314(1),17 0.69373383(72),18 0.693733124922(2)19
square octagon, bathroom tile, 4-8, truncated square (4, 82)	3	-	0.729,20 0.729724(3),21 0.7297232(5)22	0.6768,23 0.67680232(63),24 0.6768031269(6),25 0.6768031243900113(3),26
honeycomb (63)	3	3	0.6962(6),27 0.697040230(5),28 0.6970402(1),29 0.6970413(10),30 0.697043(3),31	0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=032
kagome (3, 6, 3, 6)	4	4	0.652703645... = 1 − 2 sin(π/18)33	0.5244053(3),34 0.52440516(10),35 0.52440499(2),36 0.524404978(5),37 0.52440572...,38 0.52440500(1),39 0.524404999173(3),4041 0.524404999167439(4)42 0.52440499916744820(1)43
ruby,44 rhombitrihexagonal (3, 4, 6, 4)	4	4	0.620,45 0.621819(3),46 0.62181207(7)47	0.52483258(53),48 0.5248311(1),49 0.524831461573(1)50
square (44)	4	4	0.59274(10),51 0.59274605079210(2),52 0.59274601(2),53 0.59274605095(15),54 0.59274621(13),55 0.592746050786(3),56 0.59274621(33),57 0.59274598(4),5859 0.59274605(3),60 0.593(1),61 0.591(1),62 0.569(13),63 0.59274(5)64	1⁄2
snub hexagonal, maple leaf65 (34,6)	5	5	0.57966 0.579498(3)67	0.43430621(50),68 0.43432764(3),69 0.4343283172240(6),70
snub square, puzzle (32, 4, 3, 4 )	5	5	0.550,7172 0.550806(3)73	0.41413743(46),74 0.4141378476(7),75 0.4141378565917(1),76
frieze, elongated triangular(33, 42)	5	5	0.549,77 0.550213(3),78 0.5502(8)79	0.4196(6),80 0.41964191(43),81 0.41964044(1),82 0.41964035886369(2) 83
triangular (36)	6	6	1⁄2	0.347296355... = 2 sin (π/18), 1 + p3 − 3p = 084

Note: sometimes "hexagonal" is used in place of honeycomb, although in some contexts a triangular lattice is also called a hexagonal lattice. z = bulk coordination number.

2D lattices with extended and complex neighborhoods

In this section, sq-1,2,3 corresponds to square (NN+2NN+3NN),⁸⁵ etc. Equivalent to square-2N+3N+4N,⁸⁶ sq(1,2,3).⁸⁷ tri = triangular, hc = honeycomb.

Lattice	z	Site percolation threshold	Bond percolation threshold
sq-1, sq-2, sq-3, sq-5	4	0.5927...8889 (square site)
sq-1,2, sq-2,3, sq-3,5	8	0.407...909192 (square matching)	0.25036834(6),93 0.2503685,94 0.25036840(4)95
sq-1,3	8	0.3379697	0.221499598
sq-2,5: 2NN+5NN	8	0.33799
hc-1,2,3: honeycomb-NN+2NN+3NN	12	0.300,100 0.300,101 0.302960... = 1-pc(site, hc) 102
tri-1,2: triangular-NN+2NN	12	0.295,103 0.289,104 0.290258(19)105
tri-2,3: triangular-2NN+3NN	12	0.232020(36),106 0.232020(20)107
sq-4: square-4NN	8	0.270...108
sq-1,5: square-NN+5NN (r ≤ 2)	8	0.277109
sq-1,2,3: square-NN+2NN+3NN	12	0.292,110 0.290(5) 111 0.289,112 0.288,113114	0.1522203115
sq-2,3,5: square-2NN+3NN+5NN	12	0.288116
sq-1,4: square-NN+4NN	12	0.236117
sq-2,4: square-2NN+4NN	12	0.225118
tri-4: triangular-4NN	12	0.192450(36),119 0.1924428(50)120
hc-2,4: honeycomb-2NN+4NN	12	0.2374121
tri-1,3: triangular-NN+3NN	12	0.264539(21)122
tri-1,2,3: triangular-NN+2NN+3NN	18	0.225,123 0.215,124 0.215459(36)125 0.2154657(17)126
sq-3,4: 3NN+4NN	12	0.221127
sq-1,2,5: NN+2NN+5NN	12	0.240128	0.13805374129
sq-1,3,5: NN+3NN+5NN	12	0.233130
sq-4,5: 4NN+5NN	12	0.199131
sq-1,2,4: NN+2NN+4NN	16	0.219132
sq-1,3,4: NN+3NN+4NN	16	0.208133
sq-2,3,4: 2NN+3NN+4NN	16	0.202134
sq-1,4,5: NN+4NN+5NN	16	0.187135
sq-2,4,5: 2NN+4NN+5NN	16	0.182136
sq-3,4,5: 3NN+4NN+5NN	16	0.179137
sq-1,2,3,5 asterisk pattern	16	0.208138	0.1032177139
tri-4,5: 4NN+5NN	18	0.140250(36),140
sq-1,2,3,4: NN+2NN+3NN+4NN ( r ≤ 5 {\displaystyle r\leq {\sqrt {5}}} )	20	0.19671(9),141 0.196,142 0.196724(10)143	0.0841509144
sq-1,2,4,5: NN+2NN+4NN+5NN	20	0.177145
sq-1,3,4,5: NN+3NN+4NN+5NN	20	0.172146
sq-2,3,4,5: 2NN+3NN+4NN+5NN	20	0.167147
sq-1,2,3,5,6 asterisk pattern	20		0.0783110148
sq-1,2,3,4,5: NN+2NN+3NN+4NN+5NN ( r ≤ 8 {\displaystyle r\leq {\sqrt {8}}} )	24	0.164149
tri-1,4,5: NN+4NN+5NN	24	0.131660(36)150
sq-1,...,6: NN+...+6NN (r≤3)	28	0.142151	0.0558493152
tri-2,3,4,5: 2NN+3NN+4NN+5NN	30	0.117460(36)153 0.135823(27)154
tri-1,2,3,4,5: NN+2NN+3NN+4NN+5NN	36	0.115,155 0.115740(36),156 0.1157399(58) 157
sq-1,...,7: NN+...+7NN ( r ≤ 10 {\displaystyle r\leq {\sqrt {10}}} )	36	0.113158	0.04169608159
sq lat, diamond boundary: dist. ≤ 4	40	0.105(5)160
sq-1,...,8: NN+..+8NN ( r ≤ 13 {\displaystyle r\leq {\sqrt {13}}} )	44	0.095,161 0.095765(5),162 0.09580(2)163
sq-1,...,9: NN+..+9NN (r≤4)	48	0.086164	0.02974268165
sq-1,...,11: NN+...+11NN ( r ≤ 18 {\displaystyle r\leq {\sqrt {18}}} )	60		0.02301190(3)166
sq-1,...,23 (r ≤ 7)	148		0.008342595167
sq-1,...,32: NN+...+32NN ( r ≤ 72 {\displaystyle r\leq {\sqrt {72}}} )	224		0.0053050415(33)168
sq-1,...,86: NN+...+86NN (r≤15)	708		0.001557644(4)169
sq-1,...,141: NN+...+141NN ( r ≤ 389 {\displaystyle r\leq {\sqrt {389}}} )	1224		0.000880188(90)170
sq-1,...,185: NN+...+185NN (r≤23)	1652		0.000645458(4)171
sq-1,...,317: NN+...+317NN (r≤31)	3000		0.000349601(3)172
sq-1,...,413: NN+...+413NN ( r ≤ 1280 {\displaystyle r\leq {\sqrt {1280}}} )	4016		0.0002594722(11)173
sq lat, diamond boundary: dist. ≤ 6	84	0.049(5)174
sq lat, diamond boundary: dist. ≤ 8	144	0.028(5)175
sq lat, diamond boundary: dist. ≤ 10	220	0.019(5)176
2x2 touching lattice squares* (same as sq-1,2,3,4)	20	φc = 0.58365(2),177 pc = 0.196724(10),178 0.19671(9),179
3x3 touching lattice squares* (same as sq-1,...,8))	44	φc = 0.59586(2),180 pc = 0.095765(5),181 0.09580(2) 182
4x4 touching lattice squares*	76	φc = 0.60648(1),183 pc = 0.0566227(15),184 0.05665(3),185
5x5 touching lattice squares*	116	φc = 0.61467(2),186 pc = 0.037428(2),187 0.03745(2),188
6x6 touching lattice squares*	220	pc = 0.02663(1),189
10x10 touching lattice squares*	436	φc = 0.63609(2),190 pc = 0.0100576(5) 191
within 11 x 11 square (r=5)	120		0.01048079(6)192
within 15 x 15 square (r=7)	224		0.005287692(22)193
20x20 touching lattice squares*	1676	φc = 0.65006(2),194 pc = 0.0026215(3) 195
within 31 x 31 square (r=15)	960		0.001131082(5) 196
100x100 touching lattice squares*	40396	φc = 0.66318(2),197 pc = 0.000108815(12) 198
1000x1000 touching lattice squares*	4003996	φc = 0.66639(1),199 pc = 1.09778(6)E-06 200

Here NN = nearest neighbor, 2NN = second nearest neighbor (or next nearest neighbor), 3NN = third nearest neighbor (or next-next nearest neighbor), etc. These are also called 2N, 3N, 4N respectively in some papers.²⁰¹

• For overlapping or touching squares, p c {\displaystyle p_{c}} (site) given here is the net fraction of sites occupied ϕ c {\displaystyle \phi _{c}} similar to the ϕ c {\displaystyle \phi _{c}} in continuum percolation. The case of a 2×2 square is equivalent to percolation of a square lattice NN+2NN+3NN+4NN or sq-1,2,3,4 with threshold 1 − ( 1 − ϕ c ) 1 / 4 = 0.196724 ( 10 ) … {\displaystyle 1-(1-\phi _{c})^{1/4}=0.196724(10)\ldots } with ϕ c = 0.58365 ( 2 ) {\displaystyle \phi _{c}=0.58365(2)} .²⁰² The 3×3 square corresponds to sq-1,2,3,4,5,6,7,8 with z=44 and p c = 1 − ( 1 − ϕ c ) 1 / 9 = 0.095765 ( 5 ) … {\displaystyle p_{c}=1-(1-\phi _{c})^{1/9}=0.095765(5)\ldots } . The value of z for a k x k square is (2k+1)2-5.

2D distorted lattices

Here, one distorts a regular lattice of unit spacing by moving vertices uniformly within the box ( x − α , x + α ) , ( y − α , y + α ) {\displaystyle (x-\alpha ,x+\alpha ),(y-\alpha ,y+\alpha )} , and considers percolation when sites are within Euclidean distance d {\displaystyle d} of each other.

Lattice	z ¯ {\displaystyle {\overline {z}}}	α {\displaystyle \alpha }	d {\displaystyle d}	Site percolation threshold	Bond percolation threshold
square		0.2	1.1	0.8025(2)203
		0.2	1.2	0.6667(5)204
		0.1	1.1	0.6619(1)205

Overlapping shapes on 2D lattices

Site threshold is number of overlapping objects per lattice site. k is the length (net area). Overlapping squares are shown in the complex neighborhood section. Here z is the coordination number to k-mers of either orientation, with z = k 2 + 10 k − 2 {\displaystyle z=k^{2}+10k-2} for 1 × k {\displaystyle 1\times k} sticks.

System	k	z	Site coverage φc	Site percolation threshold pc
1 x 2 dimer, square lattice	2	22	0.54691206 0.5483(2)207	0.17956(3)208 0.18019(9)209
1 x 2 aligned dimer, square lattice	2	14	0.5715(18)210	0.3454(13) 211
1 x 3 trimer, square lattice	3	37	0.49898212 0.50004(64)213	0.10880(2)214 0.1093(2)215
1 x 4 stick, square lattice	4	54	0.45761216	0.07362(2)217
1 x 5 stick, square lattice	5	73	0.42241218	0.05341(1)219
1 x 6 stick, square lattice	6	94	0.39219220	0.04063(2)221

The coverage is calculated from p c {\displaystyle p_{c}} by ϕ c = 1 − ( 1 − p c ) 2 k {\displaystyle \phi _{c}=1-(1-p_{c})^{2k}} for 1 × k {\displaystyle 1\times k} sticks, because there are 2 k {\displaystyle 2k} sites where a stick will cause an overlap with a given site.

For aligned 1 × k {\displaystyle 1\times k} sticks: ϕ c = 1 − ( 1 − p c ) k {\displaystyle \phi _{c}=1-(1-p_{c})^{k}}

Approximate formulas for thresholds of Archimedean lattices

Lattice	z	Site percolation threshold	Bond percolation threshold
(3, 122 )	3
(4, 6, 12)	3
(4, 82)	3		0.676835..., 4p3 + 3p4 − 6 p5 − 2 p6 = 1222
honeycomb (63)	3
kagome (3, 6, 3, 6)	4		0.524430..., 3p2 + 6p3 − 12 p4+ 6 p5 − p6 = 1223
(3, 4, 6, 4)	4
square (44)	4		1⁄2 (exact)
(34,6 )	5		0.434371..., 12p3 + 36p4 − 21p5 − 327 p6 + 69p7 + 2532p8 − 6533 p9 + 8256 p10 − 6255p11 + 2951p12 − 837 p13 + 126 p14 − 7p15 = 1
snub square, puzzle (32, 4, 3, 4 )	5
(33, 42)	5
triangular (36)	6	1⁄2 (exact)

AB percolation and colored percolation in 2D

In AB percolation, a p s i t e {\displaystyle p_{\mathrm {site} }} is the proportion of A sites among B sites, and bonds are drawn between sites of opposite species.²²⁴ It is also called antipercolation.

In colored percolation, occupied sites are assigned one of n {\displaystyle n} colors with equal probability, and connection is made along bonds between neighbors of different colors.²²⁵

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold
triangular AB	6	6	0.2145,226 0.21524(34),227 0.21564(3)228
AB on square-covering lattice	6	6	1 − 1 − p c ( s i t e , s q ) = 0.361835 {\displaystyle 1-{\sqrt {1-p_{c}(site,sq)}}=0.361835} 229
square three-color	4	4	0.80745(5)230
square four-color	4	4	0.73415(4)231
square five-color	4	4	0.69864(7)232
square six-color	4	4	0.67751(5)233
triangular two-color	6	6	0.72890(4)234
triangular three-color	6	6	0.63005(4)235
triangular four-color	6	6	0.59092(3)236
triangular five-color	6	6	0.56991(5)237
triangular six-color	6	6	0.55679(5)238

Site-bond percolation in 2D

Site bond percolation. Here p s {\displaystyle p_{s}} is the site occupation probability and p b {\displaystyle p_{b}} is the bond occupation probability, and connectivity is made only if both the sites and bonds along a path are occupied. The criticality condition becomes a curve f ( p s , p b ) {\displaystyle f(p_{s},p_{b})} = 0, and some specific critical pairs ( p s , p b ) {\displaystyle (p_{s},p_{b})} are listed below.

Square lattice:

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
square	4	4	0.615185(15)239	0.95
			0.667280(15)240	0.85
			0.732100(15)241	0.75
			0.75	0.726195(15)242
			0.815560(15)243	0.65
			0.85	0.615810(30)244
			0.95	0.533620(15)245

Honeycomb (hexagonal) lattice:

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
honeycomb	3	3	0.7275(5)246	0.95
			0. 0.7610(5)247	0.90
			0.7986(5)248	0.85
			0.80	0.8481(5)249
			0.8401(5)250	0.80
			0.85	0.7890(5)251
			0.90	0.7377(5)252
			0.95	0.6926(5)253

Kagome lattice:

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
kagome	4	4	0.6711(4),254 0.67097(3)255	0.95
			0.6914(5),256 0.69210(2)257	0.90
			0.7162(5),258 0.71626(3)259	0.85
			0.7428(5),260 0.74339(3)261	0.80
			0.75	0.7894(9)262
			0.7757(8),263 0.77556(3)264	0.75
			0.80	0.7152(7)265
			0.81206(3)266	0.70
			0.85	0.6556(6)267
			0.85519(3)268	0.65
			0.90	0.6046(5)269
			0.90546(3)270	0.60
			0.95	0.5615(4)271
			0.96604(4)272	0.55
			0.9854(3)273	0.53

* For values on different lattices, see "An investigation of site-bond percolation on many lattices".²⁷⁴

Approximate formula for site-bond percolation on a honeycomb lattice

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Threshold	Notes
(63) honeycomb	3	3	p b p s [ 1 − ( p b c / ( 3 − p b c ) ) ( p b − p b c ) ] = p b c {\displaystyle p_{b}p_{s}[1-({\sqrt {p_{bc}}}/(3-p_{bc}))({\sqrt {p_{b}}}-{\sqrt {p_{bc}}})]=p_{bc}} , When equal: ps = pb = 0.82199	approximate formula, ps = site prob., pb = bond prob., pbc = 1 − 2 sin (π/18),275 exact at ps=1, pb=pbc.

Archimedean duals (Laves lattices)

Laves lattices are the duals to the Archimedean lattices. Drawings from.²⁷⁶ See also Uniform tilings.

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
Cairo pentagonal D(32,4,3,4)=(2⁄3)(53)+(1⁄3)(54)	3,4	3 1⁄3	0.6501834(2),277 0.650184(5)278	0.585863... = 1 − pcbond(32,4,3,4)
Pentagonal D(33,42)=(1⁄3)(54)+(2⁄3)(53)	3,4	3 1⁄3	0.6470471(2),279 0.647084(5),280 0.6471(6)281	0.580358... = 1 − pcbond(33,42), 0.5800(6)282
D(34,6)=(1⁄5)(46)+(4⁄5)(43)	3,6	3 3⁄5	0.639447283	0.565694... = 1 − pcbond(34,6 )
dice, rhombille tiling D(3,6,3,6) = (1⁄3)(46) + (2⁄3)(43)	3,6	4	0.5851(4),284 0.585040(5)285	0.475595... = 1 − pcbond(3,6,3,6 )
ruby dual D(3,4,6,4) = (1⁄6)(46) + (2⁄6)(43) + (3⁄6)(44)	3,4,6	4	0.582410(5)286	0.475167... = 1 − pcbond(3,4,6,4 )
union jack, tetrakis square tiling D(4,82) = (1⁄2)(34) + (1⁄2)(38)	4,8	6	1⁄2	0.323197... = 1 − pcbond(4,82 )
bisected hexagon,287 cross dual D(4,6,12)= (1⁄6)(312)+(2⁄6)(36)+(1⁄2)(34)	4,6,12	6	1⁄2	0.306266... = 1 − pcbond(4,6,12)
asanoha (hemp leaf)288 D(3, 122)=(2⁄3)(33)+(1⁄3)(312)	3,12	6	1⁄2	0.259579... = 1 − pcbond(3, 122)

2-uniform lattices

Top 3 lattices: #13 #12 #36 Bottom 3 lattices: #34 #37 #11

²⁸⁹

Top 2 lattices: #35 #30 Bottom 2 lattices: #41 #42

²⁹⁰

Top 4 lattices: #22 #23 #21 #20 Bottom 3 lattices: #16 #17 #15

²⁹¹

Top 2 lattices: #31 #32 Bottom lattice: #33

²⁹²

#	Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
41	(1⁄2)(3,4,3,12) + (1⁄2)(3, 122)	4,3	3.5	0.7680(2)293	0.67493252(36)
42	(1⁄3)(3,4,6,4) + (2⁄3)(4,6,12)	4,3	31⁄3	0.7157(2)294	0.64536587(40)
36	(1⁄7)(36) + (6⁄7)(32,4,12)	6,4	4 2⁄7	0.6808(2)295	0.55778329(40)
15	(2⁄3)(32,62) + (1⁄3)(3,6,3,6)	4,4	4	0.6499(2)296	0.53632487(40)
34	(1⁄7)(36) + (6⁄7)(32,62)	6,4	4 2⁄7	0.6329(2)297	0.51707873(70)
16	(4⁄5)(3,42,6) + (1⁄5)(3,6,3,6)	4,4	4	0.6286(2)298	0.51891529(35)
17	(4⁄5)(3,42,6) + (1⁄5)(3,6,3,6)*	4,4	4	0.6279(2)299	0.51769462(35)
35	(2⁄3)(3,42,6) + (1⁄3)(3,4,6,4)	4,4	4	0.6221(2)300	0.51973831(40)
11	(1⁄2)(34,6) + (1⁄2)(32,62)	5,4	4.5	0.6171(2)301	0.48921280(37)
37	(1⁄2)(33,42) + (1⁄2)(3,4,6,4)	5,4	4.5	0.5885(2)302	0.47229486(38)
30	(1⁄2)(32,4,3,4) + (1⁄2)(3,4,6,4)	5,4	4.5	0.5883(2)303	0.46573078(72)
23	(1⁄2)(33,42) + (1⁄2)(44)	5,4	4.5	0.5720(2)304	0.45844622(40)
22	(2⁄3)(33,42) + (1⁄3)(44)	5,4	4 2⁄3	0.5648(2)305	0.44528611(40)
12	(1⁄4)(36) + (3⁄4)(34,6)	6,5	5 1⁄4	0.5607(2)306	0.41109890(37)
33	(1⁄2)(33,42) + (1⁄2)(32,4,3,4)	5,5	5	0.5505(2)307	0.41628021(35)
32	(1⁄3)(33,42) + (2⁄3)(32,4,3,4)	5,5	5	0.5504(2)308	0.41549285(36)
31	(1⁄7)(36) + (6⁄7)(32,4,3,4)	6,5	5 1⁄7	0.5440(2)309	0.40379585(40)
13	(1⁄2)(36) + (1⁄2)(34,6)	6,5	5.5	0.5407(2)310	0.38914898(35)
21	(1⁄3)(36) + (2⁄3)(33,42)	6,5	5 1⁄3	0.5342(2)311	0.39491996(40)
20	(1⁄2)(36) + (1⁄2)(33,42)	6,5	5.5	0.5258(2)312	0.38285085(38)

Inhomogeneous 2-uniform lattice

This figure shows something similar to the 2-uniform lattice #37, except the polygons are not all regular—there is a rectangle in the place of the two squares—and the size of the polygons is changed. This lattice is in the isoradial representation in which each polygon is inscribed in a circle of unit radius. The two squares in the 2-uniform lattice must now be represented as a single rectangle in order to satisfy the isoradial condition. The lattice is shown by black edges, and the dual lattice by red dashed lines. The green circles show the isoradial constraint on both the original and dual lattices. The yellow polygons highlight the three types of polygons on the lattice, and the pink polygons highlight the two types of polygons on the dual lattice. The lattice has vertex types (1⁄2)(33,42) + (1⁄2)(3,4,6,4), while the dual lattice has vertex types (1⁄15)(46)+(6⁄15)(42,52)+(2⁄15)(53)+(6⁄15)(52,4). The critical point is where the longer bonds (on both the lattice and dual lattice) have occupation probability p = 2 sin (π/18) = 0.347296... which is the bond percolation threshold on a triangular lattice, and the shorter bonds have occupation probability 1 − 2 sin(π/18) = 0.652703..., which is the bond percolation on a hexagonal lattice. These results follow from the isoradial condition³¹³ but also follow from applying the star-triangle transformation to certain stars on the honeycomb lattice. Finally, it can be generalized to having three different probabilities in the three different directions, p1, p2 and p3 for the long bonds, and 1 − p1, 1 − p2, and 1 − p3 for the short bonds, where p1, p2 and p3 satisfy the critical surface for the inhomogeneous triangular lattice.

Thresholds on 2D bow-tie and martini lattices

To the left, center, and right are: the martini lattice, the martini-A lattice, the martini-B lattice. Below: the martini covering/medial lattice, same as the 2×2, 1×1 subnet for kagome-type lattices (removed).

Some other examples of generalized bow-tie lattices (a-d) and the duals of the lattices (e-h):

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
martini (3⁄4)(3,92)+(1⁄4)(93)	3	3	0.764826..., 1 + p4 − 3p3 = 0314	0.707107... = 1/√2315
bow-tie (c)	3,4	3 1⁄7		0.672929..., 1 − 2p3 − 2p4 − 2p5 − 7p6 + 18p7 + 11p8 − 35p9 + 21p10 − 4p11 = 0316
bow-tie (d)	3,4	3 1⁄3		0.625457..., 1 − 2p2 − 3p3 + 4p4 − p5 = 0317
martini-A (2⁄3)(3,72)+(1⁄3)(3,73)	3,4	3 1⁄3	1/√2318	0.625457..., 1 − 2p2 − 3p3 + 4p4 − p5 = 0319
bow-tie dual (e)	3,4	3 2⁄3		0.595482..., 1-pcbond (bow-tie (a))320
bow-tie (b)	3,4,6	3 2⁄3		0.533213..., 1 − p − 2p3 -4p4-4p5+156+ 13p7-36p8+19p9+ p10 + p11=0321
martini covering/medial (1⁄2)(33,9) + (1⁄2)(3,9,3,9)	4	4	0.707107... = 1/√2322	0.57086651(33)
martini-B (1⁄2)(3,5,3,52) + (1⁄2)(3,52)	3, 5	4	0.618034... = 2/(1 + √5), 1- p2 − p = 0323324	1⁄2325326
bow-tie dual (f)	3,4,8	4 2⁄5		0.466787..., 1 − pcbond (bow-tie (b))327
bow-tie (a) (1⁄2)(32,4,32,4) + (1⁄2)(3,4,3)	4,6	5	0.5472(2),328 0.5479148(7)329	0.404518..., 1 − p − 6p2 + 6p3 − p5 = 0330331
bow-tie dual (h)	3,6,8	5		0.374543..., 1 − pcbond(bow-tie (d))332
bow-tie dual (g)	3,6,10	5 1⁄2	0.547... = pcsite(bow-tie(a))	0.327071..., 1 − pcbond(bow-tie (c))333
martini dual (1⁄2)(33) + (1⁄2)(39)	3,9	6	1⁄2	0.292893... = 1 − 1/√2334

Thresholds on 2D covering, medial, and matching lattices

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
(4, 6, 12) covering/medial	4	4	pcbond(4, 6, 12) = 0.693731...	0.5593140(2),335 0.559315(1)
(4, 82) covering/medial, square kagome	4	4	pcbond(4,82) = 0.676803...	0.544798017(4),336 0.54479793(34)
(34, 6) medial	4	4		0.5247495(5)337
(3,4,6,4) medial	4	4		0.51276338
(32, 4, 3, 4) medial	4	4		0.512682929(8)339
(33, 42) medial	4	4		0.5125245984(9)340
square covering (non-planar)	6	6	1⁄2	0.3371(1)341
square matching lattice (non-planar)	8	8	1 − pcsite(square) = 0.407253...	0.25036834(6)342

Thresholds on 2D chimera non-planar lattices

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
K(2,2)	4	4	0.51253(14)343	0.44778(15)344
K(3,3)	6	6	0.43760(15)345	0.35502(15)346
K(4,4)	8	8	0.38675(7)347	0.29427(12)348
K(5,5)	10	10	0.35115(13)349	0.25159(13)350
K(6,6)	12	12	0.32232(13)351	0.21942(11)352
K(7,7)	14	14	0.30052(14)353	0.19475(9)354
K(8,8)	16	16	0.28103(11)355	0.17496(10)356

Thresholds on subnet lattices

The 2 x 2, 3 x 3, and 4 x 4 subnet kagome lattices. The 2 × 2 subnet is also known as the "triangular kagome" lattice.³⁵⁷

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
checkerboard – 2 × 2 subnet	4,3			0.596303(1)358
checkerboard – 4 × 4 subnet	4,3			0.633685(9)359
checkerboard – 8 × 8 subnet	4,3			0.642318(5)360
checkerboard – 16 × 16 subnet	4,3			0.64237(1)361
checkerboard – 32 × 32 subnet	4,3			0.64219(2)362
checkerboard – ∞ {\displaystyle \infty } subnet	4,3			0.642216(10)363
kagome – 2 × 2 subnet = (3, 122) covering/medial	4		pcbond (3, 122) = 0.74042077...	0.600861966960(2),364 0.6008624(10),365 0.60086193(3)366
kagome – 3 × 3 subnet	4			0.6193296(10),367 0.61933176(5),368 0.61933044(32)
kagome – 4 × 4 subnet	4			0.625365(3),369 0.62536424(7)370
kagome – ∞ {\displaystyle \infty } subnet	4			0.628961(2)371
kagome – (1 × 1):(2 × 2) subnet = martini covering/medial	4		pcbond(martini) = 1/√2 = 0.707107...	0.57086648(36)
kagome – (1 × 1):(3 × 3) subnet	4,3		0.728355596425196...372	0.58609776(37)
kagome – (1 × 1):(4 × 4) subnet			0.738348473943256...373
kagome – (1 × 1):(5 × 5) subnet			0.743548682503071...374
kagome – (1 × 1):(6 × 6) subnet			0.746418147634282...375
kagome – (2 × 2):(3 × 3) subnet				0.61091770(30)
triangular – 2 × 2 subnet	6,4			0.471628788376
triangular – 3 × 3 subnet	6,4			0.509077793377
triangular – 4 × 4 subnet	6,4			0.524364822378
triangular – 5 × 5 subnet	6,4			0.5315976(10)379
triangular – ∞ {\displaystyle \infty } subnet	6,4			0.53993(1)380

Thresholds of random sequentially adsorbed objects

(For more results and comparison to the jamming density, see Random sequential adsorption)

system	z	Site threshold
dimers on a honeycomb lattice	3	0.69,381 0.6653 382
dimers on a triangular lattice	6	0.4872(8),383 0.4873,384
aligned linear dimers on a triangular lattice	6	0.5157(2) 385
aligned linear 4-mers on a triangular lattice	6	0.5220(2)386
aligned linear 8-mers on a triangular lattice	6	0.5281(5)387
aligned linear 12-mers on a triangular lattice	6	0.5298(8)388
linear 16-mers on a triangular lattice	6	aligned 0.5328(7)389
linear 32-mers on a triangular lattice	6	aligned 0.5407(6)390
linear 64-mers on a triangular lattice	6	aligned 0.5455(4)391
aligned linear 80-mers on a triangular lattice	6	0.5500(6)392
aligned linear k ⟶ ∞ {\displaystyle \longrightarrow \infty } on a triangular lattice	6	0.582(9)393
dimers and 5% impurities, triangular lattice	6	0.4832(7)394
parallel dimers on a square lattice	4	0.5863395
dimers on a square lattice	4	0.5617,396 0.5618(1),397 0.562,398 0.5713399
linear 3-mers on a square lattice	4	0.528400
3-site 120° angle, 5% impurities, triangular lattice	6	0.4574(9)401
3-site triangles, 5% impurities, triangular lattice	6	0.5222(9)402
linear trimers and 5% impurities, triangular lattice	6	0.4603(8)403
linear 4-mers on a square lattice	4	0.504404
linear 5-mers on a square lattice	4	0.490405
linear 6-mers on a square lattice	4	0.479406
linear 8-mers on a square lattice	4	0.474,407 0.4697(1)408
linear 10-mers on a square lattice	4	0.469409
linear 16-mers on a square lattice	4	0.4639(1)410
linear 32-mers on a square lattice	4	0.4747(2)411

The threshold gives the fraction of sites occupied by the objects when site percolation first takes place (not at full jamming). For longer k-mers see Ref.⁴¹²

Thresholds of full dimer coverings of two dimensional lattices

Here, we are dealing with networks that are obtained by covering a lattice with dimers, and then consider bond percolation on the remaining bonds. In discrete mathematics, this problem is known as the 'perfect matching' or the 'dimer covering' problem.

system	z	Bond threshold
Parallel covering, square lattice	6	0.381966...413
Shifted covering, square lattice	6	0.347296...414
Staggered covering, square lattice	6	0.376825(2)415
Random covering, square lattice	6	0.367713(2)416
Parallel covering, triangular lattice	10	0.237418...417
Staggered covering, triangular lattice	10	0.237497(2)418
Random covering, triangular lattice	10	0.235340(1)419

Thresholds of polymers (random walks) on a square lattice

System is composed of ordinary (non-avoiding) random walks of length l on the square lattice.⁴²⁰

l (polymer length)	z	Bond percolation
1	4	0.5(exact)421
2	4	0.47697(4)422
4	4	0.44892(6)423
8	4	0.41880(4)424

Thresholds of self-avoiding walks of length k added by random sequential adsorption

k	z	Site thresholds	Bond thresholds
1	4	0.593(2)425	0.5009(2)426
2	4	0.564(2)427	0.4859(2)428
3	4	0.552(2)429	0.4732(2)430
4	4	0.542(2)431	0.4630(2)432
5	4	0.531(2)433	0.4565(2)434
6	4	0.522(2)435	0.4497(2)436
7	4	0.511(2)437	0.4423(2)438
8	4	0.502(2)439	0.4348(2)440
9	4	0.493(2)441	0.4291(2)442
10	4	0.488(2)443	0.4232(2)444
11	4	0.482(2)445	0.4159(2)446
12	4	0.476(2)447	0.4114(2)448
13	4	0.471(2)449	0.4061(2)450
14	4	0.467(2)451	0.4011(2)452
15	4	0.4011(2)453	0.3979(2)454

Thresholds on 2D inhomogeneous lattices

Lattice	z	Site percolation threshold	Bond percolation threshold
bow-tie with p = 1⁄2 on one non-diagonal bond	3		0.3819654(5),455 ( 3 − 5 ) / 2 {\displaystyle (3-{\sqrt {5}})/2} 456

Thresholds for 2D continuum models

System	Φc	ηc	nc
Disks of radius r	0.67634831(2),457 0.6763475(6),458 0.676339(4),459 0.6764(4),460 0.6766(5),461 0.676(2),462 0.679,463 0.674464 0.676,465 0.680466	1.1280867(5),467 1.1276(9),468 1.12808737(6),469 1.128085(2),470 1.128059(12),471 1.13, 0.8472	1.43632505(10),473 1.43632545(8),474 1.436322(2),475 1.436289(16),476 1.436320(4),477 1.436323(3),478 1.438(2),479 1.216 (48)480
Ellipses, ε = 1.5	0.0043481	0.00431	2.059081(7)482
Ellipses, ε = 5⁄3	0.65483	1.05484	2.28485
Ellipses, ε = 2	0.6287945(12),486 0.63487	0.991000(3),488 0.99489	2.523560(8),490 2.5491
Ellipses, ε = 3	0.56492	0.82493	3.157339(8),494 3.14495
Ellipses, ε = 4	0.5496	0.69497	3.569706(8),498 3.5499
Ellipses, ε = 5	0.455,500 0.455,501 0.46502	0.607503	3.861262(12),504 3.86505
Ellipses, ε = 6			4.079365(17)506
Ellipses, ε = 7			4.249132(16)507
Ellipses, ε = 8			4.385302(15)508
Ellipses, ε = 9			4.497000(8)509
Ellipses, ε = 10	0.301,510 0.303,511 0.30512	0.358513 0.36514	4.590416(23)515 4.56,516 4.5517
Ellipses, ε = 15			4.894752(30)518
Ellipses, ε = 20	0.178,519 0.17520	0.196521	5.062313(39),522 4.99523
Ellipses, ε = 50	0.081524	0.084525	5.393863(28),526 5.38527
Ellipses, ε = 100	0.0417528	0.0426529	5.513464(40),530 5.42531
Ellipses, ε = 200	0.021532	0.0212533	5.40534
Ellipses, ε = 1000	0.0043535	0.00431	5.624756(22),536 5.5
Superellipses, ε = 1, m = 1.5	0.671537
Superellipses, ε = 2.5, m = 1.5	0.599538
Superellipses, ε = 5, m = 1.5	0.469539
Superellipses, ε = 10, m = 1.5	0.322540
disco-rectangles, ε = 1.5			1.894 541
disco-rectangles, ε = 2			2.245 542
Aligned squares of side ℓ {\displaystyle \ell }	0.66675(2),543 0.66674349(3),544 0.66653(1),545 0.6666(4),546 0.668547	1.09884280(9),548 1.0982(3),549 1.098(1)550	1.09884280(9),551 1.0982(3),552 1.098(1)553
Randomly oriented squares	0.62554075(4),554 0.6254(2)555 0.625,556	0.9822723(1),557 0.9819(6)558 0.982278(14)559	0.9822723(1),560 0.9819(6)561 0.982278(14)562
Randomly oriented squares within angle π / 4 {\displaystyle \pi /4}	0.6255(1)563	0.98216(15)564
Rectangles, ε = 1.1	0.624870(7)	0.980484(19)	1.078532(21)565
Rectangles, ε = 2	0.590635(5)	0.893147(13)	1.786294(26)566
Rectangles, ε = 3	0.5405983(34)	0.777830(7)	2.333491(22)567
Rectangles, ε = 4	0.4948145(38)	0.682830(8)	2.731318(30)568
Rectangles, ε = 5	0.4551398(31), 0.451569	0.607226(6)	3.036130(28)570
Rectangles, ε = 10	0.3233507(25), 0.319571	0.3906022(37)	3.906022(37)572
Rectangles, ε = 20	0.2048518(22)	0.2292268(27)	4.584535(54)573
Rectangles, ε = 50	0.09785513(36)	0.1029802(4)	5.149008(20)574
Rectangles, ε = 100	0.0523676(6)	0.0537886(6)	5.378856(60)575
Rectangles, ε = 200	0.02714526(34)	0.02752050(35)	5.504099(69)576
Rectangles, ε = 1000	0.00559424(6)	0.00560995(6)	5.609947(60)577
Sticks (needles) of length ℓ {\displaystyle \ell }			5.63726(2),578 5.6372858(6),579 5.637263(11),580 5.63724(18) 581
sticks with log-normal length dist. STD=0.5			4.756(3) 582
sticks with correlated angle dist. s=0.5			6.6076(4) 583
Power-law disks, x = 2.05	0.993(1)584	4.90(1)	0.0380(6)
Power-law disks, x = 2.25	0.8591(5)585	1.959(5)	0.06930(12)
Power-law disks, x = 2.5	0.7836(4)586	1.5307(17)	0.09745(11)
Power-law disks, x = 4	0.69543(6)587	1.18853(19)	0.18916(3)
Power-law disks, x = 5	0.68643(13)588	1.1597(3)	0.22149(8)
Power-law disks, x = 6	0.68241(8)589	1.1470(1)	0.24340(5)
Power-law disks, x = 7	0.6803(8)590	1.140(6)	0.25933(16)
Power-law disks, x = 8	0.67917(9)591	1.1368(5)	0.27140(7)
Power-law disks, x = 9	0.67856(12)592	1.1349(4)	0.28098(9)
Voids around disks of radius r	1 − Φc(disk) = 0.32355169(2),593 0.318(2),594 0.3261(6)595

For disks, n c = 4 r 2 N / L 2 {\displaystyle n_{c}=4r^{2}N/L^{2}} equals the critical number of disks per unit area, measured in units of the diameter 2 r {\displaystyle 2r} , where N {\displaystyle N} is the number of objects and L {\displaystyle L} is the system size

For disks, η c = π r 2 N / L 2 = ( π / 4 ) n c {\displaystyle \eta _{c}=\pi r^{2}N/L^{2}=(\pi /4)n_{c}} equals critical total disk area.

4 η c {\displaystyle 4\eta _{c}} gives the number of disk centers within the circle of influence (radius 2 r).

r c = L η c π N = L 2 n c N {\displaystyle r_{c}=L{\sqrt {\frac {\eta _{c}}{\pi N}}}={\frac {L}{2}}{\sqrt {\frac {n_{c}}{N}}}} is the critical disk radius.

η c = π a b N / L 2 {\displaystyle \eta _{c}=\pi abN/L^{2}} for ellipses of semi-major and semi-minor axes of a and b, respectively. Aspect ratio ϵ = a / b {\displaystyle \epsilon =a/b} with a > b {\displaystyle a>b} .

η c = ℓ m N / L 2 {\displaystyle \eta _{c}=\ell mN/L^{2}} for rectangles of dimensions ℓ {\displaystyle \ell } and m {\displaystyle m} . Aspect ratio ϵ = ℓ / m {\displaystyle \epsilon =\ell /m} with ℓ > m {\displaystyle \ell >m} .

η c = π x N / ( 4 L 2 ( x − 2 ) ) {\displaystyle \eta _{c}=\pi xN/(4L^{2}(x-2))} for power-law distributed disks with Prob(radius ≥ R ) = R − x {\displaystyle {\hbox{Prob(radius}}\geq R)=R^{-x}} , R ≥ 1 {\displaystyle R\geq 1} .

ϕ c = 1 − e − η c {\displaystyle \phi _{c}=1-e^{-\eta _{c}}} equals critical area fraction.

For disks, Ref.⁵⁹⁶ use ϕ c = 1 − e − π x / 2 {\displaystyle \phi _{c}=1-e^{-\pi x/2}} where x {\displaystyle x} is the density of disks of radius 1 / 2 {\displaystyle 1/{\sqrt {2}}} .

n c = ℓ 2 N / L 2 {\displaystyle n_{c}=\ell ^{2}N/L^{2}} equals number of objects of maximum length ℓ = 2 a {\displaystyle \ell =2a} per unit area.

For ellipses, n c = ( 4 ϵ / π ) η c {\displaystyle n_{c}=(4\epsilon /\pi )\eta _{c}}

For void percolation, ϕ c = e − η c {\displaystyle \phi _{c}=e^{-\eta _{c}}} is the critical void fraction.

For more ellipse values, see ⁵⁹⁷⁵⁹⁸

For more rectangle values, see ⁵⁹⁹

Both ellipses and rectangles belong to the superellipses, with | x / a | 2 m + | y / b | 2 m = 1 {\displaystyle |x/a|^{2m}+|y/b|^{2m}=1} . For more percolation values of superellipses, see.⁶⁰⁰

For the monodisperse particle systems, the percolation thresholds of concave-shaped superdisks are obtained as seen in ⁶⁰¹

For binary dispersions of disks, see ^602603604

Thresholds on 2D random and quasi-lattices

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
Relative neighborhood graph		2.5576	0.796(2)605	0.771(2)606
Voronoi tessellation	3		0.71410(2),607 0.7151*608	0.68,609 0.6670(1),610 0.6680(5),611 0.666931(5)612
Voronoi covering/medial	4		0.666931(2)613614	0.53618(2)615
Randomized kagome/square-octagon, fraction r=1⁄2	4		0.6599616
Penrose rhomb dual	4		0.6381(3)617	0.5233(2)618
Gabriel graph		4	0.6348(8),619 0.62620	0.5167(6),621 0.52622
Random-line tessellation, dual		4	0.586(2)623
Penrose rhomb		4	0.5837(3),624 0.0.5610(6) (weighted bonds)625 0.58391(1)626	0.483(5),627 0.4770(2)628
Octagonal lattice, "chemical" links (Ammann–Beenker tiling)		4	0.585629	0.48630
Octagonal lattice, "ferromagnetic" links		5.17	0.543631	0.40632
Dodecagonal lattice, "chemical" links		3.63	0.628633	0.54634
Dodecagonal lattice, "ferromagnetic" links		4.27	0.617635	0.495636
Delaunay triangulation		6	1⁄2637	0.3333(1)638 0.3326(5),639 0.333069(2)640
Uniform Infinite Planar Triangulation641		6	1⁄2	(2√3 – 1)/11 ≈ 0.2240642643

*Theoretical estimate

Thresholds on 2D correlated systems

Assuming power-law correlations C ( r ) ∼ | r | − α {\displaystyle C(r)\sim |r|^{-\alpha }}

lattice	α	Site percolation threshold	Bond percolation threshold
square	3	0.561406(4)644
square	2	0.550143(5)645
square	0.1	0.508(4)646

Thresholds on slabs

h is the thickness of the slab, h × ∞ × ∞. Boundary conditions (b.c.) refer to the top and bottom planes of the slab.

Lattice	h	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
simple cubic (open b.c.)	2	5	5	0.47424,647 0.4756648
bcc (open b.c.)	2			0.4155649
hcp (open b.c.)	2			0.2828650
diamond (open b.c.)	2			0.5451651
simple cubic (open b.c.)	3			0.4264652
bcc (open b.c.)	3			0.3531653
bcc (periodic b.c.)	3				0.21113018(38)654
hcp (open b.c.)	3			0.2548655
diamond (open b.c.)	3			0.5044656
simple cubic (open b.c.)	4			0.3997,657 0.3998658
bcc (open b.c.)	4			0.3232659
bcc (periodic b.c.)	4				0.20235168(59)660
hcp (open b.c.)	4			0.2405661
diamond (open b.c.)	4			0.4842662
simple cubic (periodic b.c.)	5	6	6		0.278102(5)663
simple cubic (open b.c.)	6			0.3708664
simple cubic (periodic b.c.)	6	6	6		0.272380(2)665
bcc (open b.c.)	6			0.2948666
hcp (open b.c.)	6			0.2261667
diamond (open b.c.)	6			0.4642668
simple cubic (periodic b.c.)	7	6	6	0.3459514(12)669	0.268459(1)670
simple cubic (open b.c.)	8			0.3557,671 0.3565672
simple cubic (periodic b.c.)	8	6	6		0.265615(5)673
bcc (open b.c.)	8			0.2811674
hcp (open b.c.)	8			0.2190675
diamond (open b.c.)	8			0.4549676
simple cubic (open b.c.)	12			0.3411677
bcc (open b.c.)	12			0.2688678
hcp (open b.c.)	12			0.2117679
diamond (open b.c.)	12			0.4456680
simple cubic (open b.c.)	16			0.3219,681 0.3339682
bcc (open b.c.)	16			0.2622683
hcp (open b.c.)	16			0.2086684
diamond (open b.c.)	16			0.4415685
simple cubic (open b.c.)	32			0.3219,686
simple cubic (open b.c.)	64			0.3165,687
simple cubic (open b.c.)	128			0.31398,688

Percolation in 3D

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	filling factor*	filling fraction*	Site percolation threshold	Bond percolation threshold
(10,3)-a oxide (or site-bond)689	23 32	2.4			0.748713(22)690	= (pc,bond(10,3) – a)1⁄2 = 0.742334(25)691
(10,3)-b oxide (or site-bond)692	23 32	2.4	0.233693	0.174	0.745317(25)694	= (pc,bond(10,3) – b)1⁄2 = 0.739388(22)695
silicon dioxide (diamond site-bond)696	4,22	2 2⁄3			0.638683(35)697
Modified (10,3)-b698	32,2	2 2⁄3				0.627699
(8,3)-a700	3	3			0.577962(33)701	0.555700(22)702
(10,3)-a703 gyroid704	3	3			0.571404(40)705	0.551060(37)706
(10,3)-b707	3	3			0.565442(40)708	0.546694(33)709
cubic oxide (cubic site-bond)710	6,23	3.5			0.524652(50)711
bcc dual		4			0.4560(6)712	0.4031(6)713
ice Ih	4	4	π √3 / 16 = 0.340087	0.147	0.433(11)714	0.388(10)715
diamond (Ice Ic)	4	4	π √3 / 16 = 0.340087	0.1462332	0.4299(8),716 0.4299870(4),717 0.426+0.08−0.02,718 0.4297(4) 719 0.4301(4),720 0.428(4),721 0.425(15),722 0.425,723724 0.436(12)725	0.3895892(5),726 0.3893(2),727 0.3893(3),728 0.388(5),729 0.3886(5),730 0.388(5)731 0.390(11)732
diamond dual		6 2⁄3			0.3904(5)733	0.2350(5)734
3D kagome (covering graph of the diamond lattice)	6		π √2 / 12 = 0.37024	0.1442	0.3895(2)735 =pc(site) for diamond dual and pc(bond) for diamond lattice736	0.2709(6)737
Bow-tie stack dual		5 1⁄3			0.3480(4)738	0.2853(4)739
honeycomb stack	5	5			0.3701(2)740	0.3093(2)741
octagonal stack dual	5	5			0.3840(4)742	0.3168(4)743
pentagonal stack		5 1⁄3			0.3394(4)744	0.2793(4)745
kagome stack	6	6	0.453450	0.1517	0.3346(4)746	0.2563(2)747
fcc dual	42,8	5 1⁄3			0.3341(5)748	0.2703(3)749
simple cubic	6	6	π / 6 = 0.5235988	0.1631574	0.307(10),750 0.307,751 0.3115(5),752 0.3116077(2),753 0.311604(6),754 0.311605(5),755 0.311600(5),756 0.3116077(4),757 0.3116081(13),758 0.3116080(4),759 0.3116060(48),760 0.3116004(35),761 0.31160768(15)762	0.247(5),763 0.2479(4),764 0.2488(2),765 0.24881182(10),766 0.2488125(25),767 0.2488126(5),768
hcp dual	44,82	5 1⁄3			0.3101(5)769	0.2573(3)770
dice stack	5,8	6	π √3 / 9 = 0.604600	0.1813	0.2998(4)771	0.2378(4)772
bow-tie stack	7	7			0.2822(6)773	0.2092(4)774
Stacked triangular / simple hexagonal	8	8			0.26240(5),775 0.2625(2),776 0.2623(2)777	0.18602(2),778 0.1859(2)779
octagonal (union-jack) stack	6,10	8			0.2524(6)780	0.1752(2)781
bcc	8	8			0.243(10),782 0.243,783 0.2459615(10),784 0.2460(3),785 0.2464(7),786 0.2458(2)787	0.178(5),788 0.1795(3),789 0.18025(15),790 0.1802875(10)791
simple cubic with 3NN (same as bcc)	8	8			0.2455(1),792 0.2457(7)793
fcc, D3	12	12	π / (3 √2) = 0.740480	0.147530	0.195,794 0.198(3),795 0.1998(6),796 0.1992365(10),797 0.19923517(20),798 0.1994(2),799 0.199236(4)800	0.1198(3),801 0.1201635(10)802 0.120169(2)803
hcp	12	12	π / (3 √2) = 0.740480	0.147545	0.195(5),804 0.1992555(10)805	0.1201640(10),806 0.119(2)807
La2−x Srx Cu O4	12	12			0.19927(2)808
simple cubic with 2NN (same as fcc)	12	12			0.1991(1)809
simple cubic with NN+4NN	12	12			0.15040(12),810 0.1503793(7)811	0.1068263(7)812
simple cubic with 3NN+4NN	14	14			0.20490(12)813	0.1012133(7)814
bcc NN+2NN (= sc(3,4) sc-3NN+4NN)	14	14			0.175,815 0.1686,(20)816 0.1759432(8)	0.0991(5),817 0.1012133(7),818 0.1759432(8) 819
Nanotube fibers on FCC	14	14			0.1533(13)820
simple cubic with NN+3NN	14	14			0.1420(1)821	0.0920213(7)822
simple cubic with 2NN+4NN	18	18			0.15950(12)823	0.0751589(9)824
simple cubic with NN+2NN	18	18			0.137,825 0.136,826 0.1372(1),827 0.13735(5), 0.1373045(5)828	0.0752326(6) 829
fcc with NN+2NN (=sc-2NN+4NN)	18	18			0.136,830 0.1361408(8)831	0.0751589(9) 832
simple cubic with short-length correlation	6+	6+			0.126(1)833
simple cubic with NN+3NN+4NN	20	20			0.11920(12)834	0.0624379(9)835
simple cubic with 2NN+3NN	20	20			0.1036(1)836	0.0629283(7)837
simple cubic with NN+2NN+4NN	24	24			0.11440(12)838	0.0533056(6)839
simple cubic with 2NN+3NN+4NN	26	26			0.11330(12)840	0.0474609(9)
simple cubic with NN+2NN+3NN	26	26			0.097,841 0.0976(1),842 0.0976445(10), 0.0976444(6)843	0.0497080(10)844
bcc with NN+2NN+3NN	26	26			0.095,845 0.0959084(6)846	0.0492760(10)847
simple cubic with NN+2NN+3NN+4NN	32	32			0.10000(12),848 0.0801171(9)849	0.0392312(8)850
fcc with NN+2NN+3NN	42	42			0.061,851 0.0610(5),852 0.0618842(8)853	0.0290193(7) 854
fcc with NN+2NN+3NN+4NN	54	54			0.0500(5)855
sc-1,2,3,4,5 simple cubic with NN+2NN+3NN+4NN+5NN	56	56			0.0461815(5)856	0.0210977(7)857
sc-1,...,6 (2x2x2 cube 858)	80	80			0.0337049(9),859 0.03373(13)860	0.0143950(10)861
sc-1,...,7	92	92			0.0290800(10)862	0.0123632(8)863
sc-1,...,8	122	122			0.0218686(6)864	0.0091337(7)865
sc-1,...,9	146	146			0.0184060(10)866	0.0075532(8)867
sc-1,...,10	170	170				0.0064352(8)868
sc-1,...,11	178	178				0.0061312(8)869
sc-1,...,12	202	202				0.0053670(10)870
sc-1,...,13	250	250				0.0042962(8)871
3x3x3 cube	274	274			φc= 0.76564(1),872 pc = 0.0098417(7),873 0.009854(6)874
4x4x4 cube	636	636			φc=0.76362(1),875 pc = 0.0042050(2),876 0.004217(3)877
5x5x5 cube	1214	1250			φc=0.76044(2),878 pc = 0.0021885(2),879 0.002185(4)880
6x6x6 cube	2056	2056			0.001289(2)881

Filling factor = fraction of space filled by touching spheres at every lattice site (for systems with uniform bond length only). Also called Atomic Packing Factor.

Filling fraction (or Critical Filling Fraction) = filling factor * pc(site).

NN = nearest neighbor, 2NN = next-nearest neighbor, 3NN = next-next-nearest neighbor, etc.

kxkxk cubes are cubes of occupied sites on a lattice, and are equivalent to extended-range percolation of a cube of length (2k+1), with edges and corners removed, with z = (2k+1)3-12(2k-1)-9 (center site not counted in z).

Question: the bond thresholds for the hcp and fcc lattice agree within the small statistical error. Are they identical, and if not, how far apart are they? Which threshold is expected to be bigger? Similarly for the ice and diamond lattices. See ⁸⁸²

System	polymer Φc
percolating excluded volume of athermal polymer matrix (bond-fluctuation model on cubic lattice)	0.4304(3)883

3D distorted lattices

Here, one distorts a regular lattice of unit spacing by moving vertices uniformly within the cube ( x − α , x + α ) , ( y − α , y + α ) , ( z − α , z + α ) {\displaystyle (x-\alpha ,x+\alpha ),(y-\alpha ,y+\alpha ),(z-\alpha ,z+\alpha )} , and considers percolation when sites are within Euclidean distance d {\displaystyle d} of each other.

Lattice	z ¯ {\displaystyle {\overline {z}}}	α {\displaystyle \alpha }	d {\displaystyle d}	Site percolation threshold	Bond percolation threshold
cubic		0.05	1.0	0.60254(3)884
		0.1	1.00625	0.58688(4)885
		0.15	1.025	0.55075(2)886
		0.175	1.05	0.50645(5)887
		0.2	1.1	0.44342(3)888

Overlapping shapes on 3D lattices

Site threshold is the number of overlapping objects per lattice site. The coverage φc is the net fraction of sites covered, and v is the volume (number of cubes). Overlapping cubes are given in the section on thresholds of 3D lattices. Here z is the coordination number to k-mers of either orientation, with z = 6 k 2 + 18 k − 4 {\displaystyle z=6k^{2}+18k-4}

System	k	z	Site coverage φc	Site percolation threshold pc
1 x 2 dimer, cubic lattice	2	56	0.24542889	0.045847(2)890
1 x 3 trimer, cubic lattice	3	104	0.19578891	0.023919(9)892
1 x 4 stick, cubic lattice	4	164	0.16055893	0.014478(7)894
1 x 5 stick, cubic lattice	5	236	0.13488895	0.009613(8)896
1 x 6 stick, cubic lattice	6	320	0.11569897	0.006807(2)898
2 x 2 plaquette, cubic lattice	2		0.22710899	0.021238(2)900
3 x 3 plaquette, cubic lattice	3		0.18686901	0.007632(5)902
4 x 4 plaquette, cubic lattice	4		0.16159903	0.003665(3)904
5 x 5 plaquette, cubic lattice	5		0.14316905	0.002058(5)906
6 x 6 plaquette, cubic lattice	6		0.12900907	0.001278(5)908

The coverage is calculated from p c {\displaystyle p_{c}} by ϕ c = 1 − ( 1 − p c ) 3 k {\displaystyle \phi _{c}=1-(1-p_{c})^{3k}} for sticks, and ϕ c = 1 − ( 1 − p c ) 3 k 2 {\displaystyle \phi _{c}=1-(1-p_{c})^{3k^{2}}} for plaquettes.

Dimer percolation in 3D

System	Site percolation threshold	Bond percolation threshold
Simple cubic		0.2555(1)909

Thresholds for 3D continuum models

All overlapping except for jammed spheres and polymer matrix.

System	Φc	ηc
Spheres of radius r	0.289,910 0.293,911 0.286,912 0.295.913 0.2895(5),914 0.28955(7),915 0.2896(7),916 0.289573(2),917 0.2896,918 0.2854,919 0.290,920 0.290,921 0.2895693(26)922	0.3418(7),923 0.3438(13),924 0.341889(3),925 0.3360,926 0.34189(2) 927 [corrected], 0.341935(8),928 0.335,929
Oblate ellipsoids with major radius r and aspect ratio 4⁄3	0.2831930	0.3328931
Prolate ellipsoids with minor radius r and aspect ratio 3⁄2	0.2757,932 0.2795,933 0.2763934	0.3278935
Oblate ellipsoids with major radius r and aspect ratio 2	0.2537,936 0.2629,937 0.254938	0.3050939
Prolate ellipsoids with minor radius r and aspect ratio 2	0.2537,940 0.2618,941 0.25(2),942 0.2507943	0.3035,944 0.29(3)945
Oblate ellipsoids with major radius r and aspect ratio 3	0.2289946	0.2599947
Prolate ellipsoids with minor radius r and aspect ratio 3	0.2033,948 0.2244,949 0.20(2)950	0.2541,951 0.22(3)952
Oblate ellipsoids with major radius r and aspect ratio 4	0.2003953	0.2235954
Prolate ellipsoids with minor radius r and aspect ratio 4	0.1901,955 0.16(2)956	0.2108,957 0.17(3)958
Oblate ellipsoids with major radius r and aspect ratio 5	0.1757959	0.1932960
Prolate ellipsoids with minor radius r and aspect ratio 5	0.1627,961 0.13(2)962	0.1776,963 0.15(2)964
Oblate ellipsoids with major radius r and aspect ratio 10	0.0895,965 0.1058966	0.1118967
Prolate ellipsoids with minor radius r and aspect ratio 10	0.0724,968 0.08703,969 0.07(2)970	0.09105,971 0.07(2)972
Oblate ellipsoids with major radius r and aspect ratio 100	0.01248973	0.01256974
Prolate ellipsoids with minor radius r and aspect ratio 100	0.006949975	0.006973976
Oblate ellipsoids with major radius r and aspect ratio 1000	0.001275977	0.001276978
Oblate ellipsoids with major radius r and aspect ratio 2000	0.000637979	0.000637980
Spherocylinders with H/D = 1	0.2439(2)981
Spherocylinders with H/D = 4	0.1345(1)982
Spherocylinders with H/D = 10	0.06418(20)983
Spherocylinders with H/D = 50	0.01440(8)984
Spherocylinders with H/D = 100	0.007156(50)985
Spherocylinders with H/D = 200	0.003724(90)986
Aligned cylinders	0.2819(2)987	0.3312(1)988
Aligned cubes of side ℓ = 2 a {\displaystyle \ell =2a}	0.2773(2)989 0.27727(2),990 0.27730261(79)991	0.3247(3),992 0.3248(3),993 0.32476(4)994 0.324766(1)995
Randomly oriented icosahedra		0.3030(5)996
Randomly oriented dodecahedra		0.2949(5)997
Randomly oriented octahedra		0.2514(6)998
Randomly oriented cubes of side ℓ = 2 a {\displaystyle \ell =2a}	0.2168(2)999 0.2174,1000	0.2444(3),1001 0.2443(5)1002
Randomly oriented tetrahedra		0.1701(7)1003
Randomly oriented disks of radius r (in 3D)		0.9614(5)1004
Randomly oriented square plates of side π r {\displaystyle {\sqrt {\pi }}r}		0.8647(6)1005
Randomly oriented triangular plates of side 2 π / 3 1 / 4 r {\displaystyle {\sqrt {2\pi }}/3^{1/4}r}		0.7295(6)1006
Jammed spheres (average z = 6)	0.183(3),1007 0.1990,1008 see also contact network of jammed spheres below.	0.59(1)1009 (volume fraction of all spheres)

η c = ( 4 / 3 ) π r 3 N / L 3 {\displaystyle \eta _{c}=(4/3)\pi r^{3}N/L^{3}} is the total volume (for spheres), where N is the number of objects and L is the system size.

ϕ c = 1 − e − η c {\displaystyle \phi _{c}=1-e^{-\eta _{c}}} is the critical volume fraction, valid for overlapping randomly placed objects.

For disks and plates, these are effective volumes and volume fractions.

For void ("Swiss-Cheese" model), ϕ c = e − η c {\displaystyle \phi _{c}=e^{-\eta _{c}}} is the critical void fraction.

For more results on void percolation around ellipsoids and elliptical plates, see.¹⁰¹⁰

For more ellipsoid percolation values see.¹⁰¹¹

For spherocylinders, H/D is the ratio of the height to the diameter of the cylinder, which is then capped by hemispheres. Additional values are given in.¹⁰¹²

For superballs, m is the deformation parameter, the percolation values are given in.,^10131014 In addition, the thresholds of concave-shaped superballs are also determined in ¹⁰¹⁵

For cuboid-like particles (superellipsoids), m is the deformation parameter, more percolation values are given in.¹⁰¹⁶

Void percolation in 3D

Void percolation refers to percolation in the space around overlapping objects. Here ϕ c {\displaystyle \phi _{c}} refers to the fraction of the space occupied by the voids (not of the particles) at the critical point, and is related to η c {\displaystyle \eta _{c}} by ϕ c = e − η c {\displaystyle \phi _{c}=e^{-\eta _{c}}} . η c {\displaystyle \eta _{c}} is defined as in the continuum percolation section above.

System	Φc	ηc
Voids around disks of radius r		22.86(2)1017
Voids around randomly oriented tetrahedra	0.0605(6)1018
Voids around oblate ellipsoids of major radius r and aspect ratio 32	0.5308(7)1019	0.63331020
Voids around oblate ellipsoids of major radius r and aspect ratio 16	0.3248(5)1021	1.1251022
Voids around oblate ellipsoids of major radius r and aspect ratio 10		1.542(1)1023
Voids around oblate ellipsoids of major radius r and aspect ratio 8	0.1615(4)1024	1.8231025
Voids around oblate ellipsoids of major radius r and aspect ratio 4	0.0711(2)1026	2.643,1027 2.618(5)1028
Voids around oblate ellipsoids of major radius r and aspect ratio 2		3.239(4) 1029
Voids around prolate ellipsoids of aspect ratio 8	0.0415(7)1030
Voids around prolate ellipsoids of aspect ratio 6	0.0397(7)1031
Voids around prolate ellipsoids of aspect ratio 4	0.0376(7)1032
Voids around prolate ellipsoids of aspect ratio 3	0.03503(50)1033
Voids around prolate ellipsoids of aspect ratio 2	0.0323(5)1034
Voids around aligned square prisms of aspect ratio 2	0.0379(5) 1035
Voids around randomly oriented square prisms of aspect ratio 20	0.0534(4) 1036
Voids around randomly oriented square prisms of aspect ratio 15	0.0535(4) 1037
Voids around randomly oriented square prisms of aspect ratio 10	0.0524(5) 1038
Voids around randomly oriented square prisms of aspect ratio 8	0.0523(6) 1039
Voids around randomly oriented square prisms of aspect ratio 7	0.0519(3) 1040
Voids around randomly oriented square prisms of aspect ratio 6	0.0519(5) 1041
Voids around randomly oriented square prisms of aspect ratio 5	0.0515(7) 1042
Voids around randomly oriented square prisms of aspect ratio 4	0.0505(7) 1043
Voids around randomly oriented square prisms of aspect ratio 3	0.0485(11) 1044
Voids around randomly oriented square prisms of aspect ratio 5/2	0.0483(8) 1045
Voids around randomly oriented square prisms of aspect ratio 2	0.0465(7) 1046
Voids around randomly oriented square prisms of aspect ratio 3/2	0.0461(14) 1047
Voids around hemispheres	0.0455(6)1048
Voids around aligned tetrahedra	0.0605(6)1049
Voids around randomly oriented tetrahedra	0.0605(6)1050
Voids around aligned cubes	0.036(1),1051 0.0381(3)1052
Voids around randomly oriented cubes	0.0452(6),1053 0.0449(5)1054
Voids around aligned octahedra	0.0407(3)1055
Voids around randomly oriented octahedra	0.0398(5)1056
Voids around aligned dodecahedra	0.0356(3)1057
Voids around randomly oriented dodecahedra	0.0360(3)1058
Voids around aligned icosahedra	0.0346(3)1059
Voids around randomly oriented icosahedra	0.0336(7)1060
Voids around spheres	0.034(7),1061 0.032(4),1062 0.030(2),1063 0.0301(3),1064 0.0294,1065 0.0300(3),1066 0.0317(4),1067 0.0308(5)1068 0.0301(1),1069 0.0301(1)1070	3.506(8),1071 3.515(6),1072 3.510(2)1073

Thresholds on 3D random and quasi-lattices

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Bond percolation threshold
Contact network of packed spheres		6	0.310(5),1074 0.287(50),1075 0.3116(3),1076
Random-plane tessellation, dual		6	0.290(7)1077
Icosahedral Penrose		6	0.2851078	0.2251079
Penrose w/2 diagonals		6.764	0.2711080	0.2071081
Penrose w/8 diagonals		12.764	0.1881082	0.1111083
Voronoi network		15.54	0.1453(20)1084	0.0822(50)1085

Thresholds for other 3D models

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold	Critical coverage fraction ϕ c {\displaystyle \phi _{c}}	Bond percolation threshold
Drilling percolation, simple cubic lattice*	6	6	0.6345(3),1086 0.6339(5),1087 0.633965(15)1088	0.25480
Drill in z direction on cubic lattice, remove single sites	6	6	0.592746 (columns), 0.4695(10) (sites)1089	0.2784
Random tube model, simple cubic lattice†					0.231456(6)1090
Pac-Man percolation, simple cubic lattice					0.139(6)1091

∗ {\displaystyle ^{*}} In drilling percolation, the site threshold p c {\displaystyle p_{c}} represents the fraction of columns in each direction that have not been removed, and ϕ c = p c 3 {\displaystyle \phi _{c}=p_{c}^{3}} . For the 1d drilling, we have ϕ c = p c {\displaystyle \phi _{c}=p_{c}} (columns) p c {\displaystyle p_{c}} (sites).

† In tube percolation, the bond threshold represents the value of the parameter μ {\displaystyle \mu } such that the probability of putting a bond between neighboring vertical tube segments is 1 − e − μ h i {\displaystyle 1-e^{-\mu h_{i}}} , where h i {\displaystyle h_{i}} is the overlap height of two adjacent tube segments.¹⁰⁹²

Thresholds in different dimensional spaces

Continuum models in higher dimensions

d	System	Φc	ηc
4	Overlapping hyperspheres	0.1223(4)1093	0.1300(13),1094 0.1304(5),1095 0.1210268(19)1096
4	Aligned hypercubes	0.1132(5),1097 0.1132348(17) 1098	0.1201(6)1099
4	Voids around hyperspheres	0.00211(2)1100	6.161(10)1101 6.248(2),1102
5	Overlapping hyperspheres		0.0544(6),1103 0.05443(7),1104 0.0522524(69)1105
5	Aligned hypercubes	0.04900(7),1106 0.0481621(13)1107	0.05024(7)1108
5	Voids around hyperspheres	1.26(6)x10−41109	8.98(4),1110 9.170(8)1111
6	Overlapping hyperspheres		0.02391(31),1112 0.02339(5)1113
6	Aligned hypercubes	0.02082(8),1114 0.0213479(10)1115	0.02104(8)1116
6	Voids around hyperspheres	8.0(6)x10−6 1117	11.74(8),1118 12.24(2),1119
7	Overlapping hyperspheres		0.01102(16),1120 0.01051(3)1121
7	Aligned hypercubes	0.00999(5),1122 0.0097754(31)1123	0.01004(5)1124
7	Voids around hyperspheres		15.46(5)1125
8	Overlapping hyperspheres		0.00516(8),1126 0.004904(6)1127
8	Aligned hypercubes		0.004498(5)1128
8	Voids around hyperspheres		18.64(8)1129
9	Overlapping hyperspheres		0.002353(4)1130
9	Aligned hypercubes		0.002166(4)1131
9	Voids around hyperspheres		22.1(4)1132
10	Overlapping hyperspheres		0.001138(3)1133
10	Aligned hypercubes		0.001058(4)1134
11	Overlapping hyperspheres		0.0005530(3)1135
11	Aligned hypercubes		0.0005160(3)1136

η c = ( π d / 2 / Γ [ d / 2 + 1 ] ) r d N / L d . {\displaystyle \eta _{c}=(\pi ^{d/2}/\Gamma [d/2+1])r^{d}N/L^{d}.}

In 4d, η c = ( 1 / 2 ) π 2 r 4 N / L 4 {\displaystyle \eta _{c}=(1/2)\pi ^{2}r^{4}N/L^{4}} .

In 5d, η c = ( 8 / 15 ) π 2 r 5 N / L 5 {\displaystyle \eta _{c}=(8/15)\pi ^{2}r^{5}N/L^{5}} .

In 6d, η c = ( 1 / 6 ) π 3 r 6 N / L 6 {\displaystyle \eta _{c}=(1/6)\pi ^{3}r^{6}N/L^{6}} .

ϕ c = 1 − e − η c {\displaystyle \phi _{c}=1-e^{-\eta _{c}}} is the critical volume fraction, valid for overlapping objects.

For void models, ϕ c = e − η c {\displaystyle \phi _{c}=e^{-\eta _{c}}} is the critical void fraction, and η c {\displaystyle \eta _{c}} is the total volume of the overlapping objects

Thresholds on hypercubic lattices

d	z	Site thresholds	Bond thresholds
4	8	0.198(1)1137 0.197(6),1138 0.1968861(14),1139 0.196889(3),1140 0.196901(5),1141 0.19680(23),1142 0.1968904(65),1143 0.19688561(3)1144	0.1600(1),1145 0.16005(15),1146 0.1601314(13),1147 0.160130(3),1148 0.1601310(10),1149 0.1601312(2),1150 0.16013122(6)1151
5	10	0.141(1),0.198(1)1152 0.141(3),1153 0.1407966(15),1154 0.1407966(26),1155 0.14079633(4)1156	0.1181(1),1157 0.118(1),1158 0.11819(4),1159 0.118172(1),1160 0.1181718(3)1161 0.11817145(3)1162
6	12	0.106(1),1163 0.108(3),1164 0.109017(2),1165 0.1090117(30),1166 0.109016661(8)1167	0.0943(1),1168 0.0942(1),1169 0.0942019(6),1170 0.09420165(2)1171
7	14	0.05950(5),1172 0.088939(20),1173 0.0889511(9),1174 0.0889511(90),1175 0.088951121(1),1176	0.0787(1),1177 0.078685(30),1178 0.0786752(3),1179 0.078675230(2)1180
8	16	0.0752101(5),1181 0.075210128(1)1182	0.06770(5),1183 0.06770839(7),1184 0.0677084181(3)1185
9	18	0.0652095(3),1186 0.0652095348(6)1187	0.05950(5),1188 0.05949601(5),1189 0.0594960034(1)1190
10	20	0.0575930(1),1191 0.0575929488(4)1192	0.05309258(4),1193 0.0530925842(2)1194
11	22	0.05158971(8),1195 0.0515896843(2)1196	0.04794969(1),1197 0.04794968373(8)1198
12	24	0.04673099(6),1199 0.0467309755(1)1200	0.04372386(1),1201 0.04372385825(10)1202
13	26	0.04271508(8),1203 0.04271507960(10)1204	0.04018762(1),1205 0.04018761703(6)1206

For thresholds on high dimensional hypercubic lattices, we have the asymptotic series expansions ^{1207 1208 1209}

p c s i t e ( d ) = σ − 1 + 3 2 σ − 2 + 15 4 σ − 3 + 83 4 σ − 4 + 6577 48 σ − 5 + 119077 96 σ − 6 + O ( σ − 7 ) {\displaystyle p_{c}^{\mathrm {site} }(d)=\sigma ^{-1}+{\frac {3}{2}}\sigma ^{-2}+{\frac {15}{4}}\sigma ^{-3}+{\frac {83}{4}}\sigma ^{-4}+{\frac {6577}{48}}\sigma ^{-5}+{\frac {119077}{96}}\sigma ^{-6}+{\mathcal {O}}(\sigma ^{-7})}

p c b o n d ( d ) = σ − 1 + 5 2 σ − 3 + 15 2 σ − 4 + 57 σ − 5 + 4855 12 σ − 6 + O ( σ − 7 ) {\displaystyle p_{c}^{\mathrm {bond} }(d)=\sigma ^{-1}+{\frac {5}{2}}\sigma ^{-3}+{\frac {15}{2}}\sigma ^{-4}+57\sigma ^{-5}+{\frac {4855}{12}}\sigma ^{-6}+{\mathcal {O}}(\sigma ^{-7})}

where σ = 2 d − 1 {\displaystyle \sigma =2d-1} . For 13-dimensional bond percolation, for example, the error with the measured value is less than 10−6, and these formulas can be useful for higher-dimensional systems.

Thresholds in other higher-dimensional lattices

d	lattice	z	Site thresholds	Bond thresholds
4	diamond	5	0.2978(2)1210	0.2715(3)1211
4	kagome	8	0.2715(3)1212	0.177(1) 1213
4	bcc	16	0.1037(3)1214	0.074(1),1215 0.074212(1)1216
4	fcc, D4, hypercubic 2NN	24	0.0842(3),1217 0.08410(23),1218 0.0842001(11)1219	0.049(1),1220 0.049517(1),1221 0.0495193(8)1222
4	hypercubic NN+2NN	32	0.06190(23),1223 0.0617731(19)1224	0.035827(1),1225 0.0338047(27)1226
4	hypercubic 3NN	32	0.04540(23)1227
4	hypercubic NN+3NN	40	0.04000(23)1228	0.0271892(22)1229
4	hypercubic 2NN+3NN	56	0.03310(23)1230	0.0194075(15)1231
4	hypercubic NN+2NN+3NN	64	0.03190(23),1232 0.0319407(13)1233	0.0171036(11)1234
4	hypercubic NN+2NN+3NN+4NN	88	0.0231538(12)1235	0.0122088(8)1236
4	hypercubic NN+...+5NN	136	0.0147918(12)1237	0.0077389(9)1238
4	hypercubic NN+...+6NN	232	0.0088400(10)1239	0.0044656(11)1240
4	hypercubic NN+...+7NN	296	0.0070006(6)1241	0.0034812(7)1242
4	hypercubic NN+...+8NN	320	0.0064681(9)1243	0.0032143(8)1244
4	hypercubic NN+...+9NN	424	0.0048301(9)1245	0.0024117(7)1246
5	diamond	6	0.2252(3)1247	0.2084(4)1248
5	kagome	10	0.2084(4)1249	0.130(2)1250
5	bcc	32	0.0446(4)1251	0.033(1)1252
5	fcc, D5, hypercubic 2NN	40	0.0431(3),1253 0.0435913(6)1254	0.026(2),1255 0.0271813(2)1256
5	hypercubic NN+2NN	50	0.0334(2)1257	0.0213(1)1258
6	diamond	7	0.1799(5)1259	0.1677(7)1260
6	kagome	12	0.1677(7)1261
6	fcc, D6	60	0.0252(5),1262 0.02602674(12)1263	0.01741556(5)1264
6	bcc	64	0.0199(5)1265
6	E61266	72	0.02194021(14)1267	0.01443205(8)1268
7	fcc, D7	84	0.01716730(5)1269	0.012217868(13)1270
7	E71271	126	0.01162306(4)1272	0.00808368(2)1273
8	fcc, D8	112	0.01215392(4)1274	0.009081804(6)1275
8	E81276	240	0.00576991(2)1277	0.004202070(2)1278
9	fcc, D9	144	0.00905870(2)1279	0.007028457(3)1280
9	Λ 9 {\displaystyle \Lambda _{9}} 1281	272	0.00480839(2)1282	0.0037006865(11)1283
10	fcc, D10	180	0.007016353(9)1284	0.005605579(6)1285
11	fcc, D11	220	0.005597592(4)1286	0.004577155(3)1287
12	fcc, D12	264	0.004571339(4)1288	0.003808960(2)1289
13	fcc, D13	312	0.003804565(3)1290	0.0032197013(14)1291

Thresholds in one-dimensional long-range percolation

In a one-dimensional chain we establish bonds between distinct sites i {\displaystyle i} and j {\displaystyle j} with probability p = C | i − j | 1 + σ {\displaystyle p={\frac {C}{|i-j|^{1+\sigma }}}} decaying as a power-law with an exponent σ > 0 {\displaystyle \sigma >0} . Percolation occurs^12921293 at a critical value C c < 1 {\displaystyle C_{c}<1} for σ < 1 {\displaystyle \sigma <1} . The numerically determined percolation thresholds are given by:¹²⁹⁴

σ {\displaystyle \sigma }	C c {\displaystyle C_{c}}	Critical thresholds C c {\displaystyle C_{c}} as a function of σ {\displaystyle \sigma } .1295The dotted line is the rigorous lower bound.1296
0.1	0.047685(8)
0.2	0.093211(16)
0.3	0.140546(17)
0.4	0.193471(15)
0.5	0.25482(5)
0.6	0.327098(6)
0.7	0.413752(14)
0.8	0.521001(14)
0.9	0.66408(7)

Thresholds on hyperbolic, hierarchical, and tree lattices

In these lattices there may be two percolation thresholds: the lower threshold is the probability above which infinite clusters appear, and the upper is the probability above which there is a unique infinite cluster.

Lattice	z	z ¯ {\displaystyle {\overline {z}}}	Site percolation threshold		Bond percolation threshold
			Lower	Upper	Lower	Upper
{3,7} hyperbolic	7	7	0.26931171(7),1297 0.201298	0.73068829(7),1299 0.73(2)1300	0.20,1301 0.1993505(5)1302	0.37,1303 0.4694754(8)1304
{3,8} hyperbolic	8	8	0.20878618(9)1305	0.79121382(9)1306	0.1601555(2)1307	0.4863559(6)1308
{3,9} hyperbolic	9	9	0.1715770(1)1309	0.8284230(1)1310	0.1355661(4)1311	0.4932908(1)1312
{4,5} hyperbolic	5	5	0.29890539(6)1313	0.8266384(5)1314	0.27,1315 0.2689195(3)1316	0.52,1317 0.6487772(3) 1318
{4,6} hyperbolic	6	6	0.22330172(3)1319	0.87290362(7)1320	0.20714787(9)1321	0.6610951(2)1322
{4,7} hyperbolic	7	7	0.17979594(1)1323	0.89897645(3)1324	0.17004767(3)1325	0.66473420(4)1326
{4,8} hyperbolic	8	8	0.151035321(9)1327	0.91607962(7)1328	0.14467876(3)1329	0.66597370(3)1330
{4,9} hyperbolic	8	8	0.13045681(3)1331	0.92820305(3)1332	0.1260724(1)1333	0.66641596(2)1334
{5,5} hyperbolic	5	5	0.26186660(5)1335	0.89883342(7)1336	0.263(10),1337 0.25416087(3)1338	0.749(10)1339 0.74583913(3)1340
{7,3} hyperbolic	3	3	0.54710885(10)1341	0.8550371(5),1342 0.86(2)1343	0.53,1344 0.551(10),1345 0.5305246(8)1346	0.72,1347 0.810(10),1348 0.8006495(5)1349
{∞,3} Cayley tree	3	3	1⁄2		1⁄21350	11351
Enhanced binary tree (EBT)					0.304(1),1352 0.306(10),1353 (√13 − 3)/2 = 0.3027761354	0.48,1355 0.564(1),1356 0.564(10),1357 1⁄21358
Enhanced binary tree dual					0.436(1),1359 0.452(10)1360	0.696(1),1361 0.699(10)1362
Non-Planar Hanoi Network (HN-NP)					0.3194451363	0.3819961364
Cayley tree with grandparents		8			0.1586563261365

Note: {m,n} is the Schläfli symbol, signifying a hyperbolic lattice in which n regular m-gons meet at every vertex

For bond percolation on {P,Q}, we have by duality p c , ℓ ( P , Q ) + p c , u ( Q , P ) = 1 {\displaystyle p_{c,\ell }(P,Q)+p_{c,u}(Q,P)=1} . For site percolation, p c , ℓ ( 3 , Q ) + p c , u ( 3 , Q ) = 1 {\displaystyle p_{c,\ell }(3,Q)+p_{c,u}(3,Q)=1} because of the self-matching of triangulated lattices.

Cayley tree (Bethe lattice) with coordination number z : p c = 1 / ( z − 1 ) {\displaystyle z:p_{c}=1/(z-1)}

Thresholds for directed percolation

Lattice	z	Site percolation threshold	Bond percolation threshold
(1+1)-d honeycomb	1.5	0.8399316(2),1366 0.839933(5),1367 = p c ( s i t e ) {\displaystyle ={\sqrt {p_{c}({\mathrm {site} })}}} of (1+1)-d sq.	0.8228569(2),1368 0.82285680(6)1369
(1+1)-d kagome	2	0.7369317(2),1370 0.73693182(4)1371	0.6589689(2),1372 0.65896910(8)1373
(1+1)-d square, diagonal	2	0.705489(4),1374 0.705489(4),1375 0.70548522(4),1376 0.70548515(20),1377 0.7054852(3),1378	0.644701(2),1379 0.644701(1),1380 0.644701(1),1381 0.6447006(10),1382 0.64470015(5),1383 0.644700185(5),1384 0.6447001(2),1385 0.643(2)1386
(1+1)-d triangular	3	0.595646(3),1387 0.5956468(5),1388 0.5956470(3)1389	0.478018(2),1390 0.478025(1),1391 0.4780250(4)1392 0.479(3)1393
(2+1)-d simple cubic, diagonal planes	3	0.43531(1),1394 0.43531411(10)1395	0.382223(7),1396 0.38222462(6)1397 0.383(3)1398
(2+1)-d square nn (= bcc)	4	0.3445736(3),1399 0.344575(15)1400 0.3445740(2)1401	0.2873383(1),1402 0.287338(3)1403 0.28733838(4)1404 0.287(3)1405
(2+1)-d fcc			0.199(2))1406
(3+1)-d hypercubic, diagonal	4	0.3025(10),1407 0.30339538(5) 1408	0.26835628(5),1409 0.2682(2)1410
(3+1)-d cubic, nn	6	0.2081040(4)1411	0.1774970(5)1412
(3+1)-d bcc	8	0.160950(30),1413 0.16096128(3)1414	0.13237417(2)1415
(4+1)-d hypercubic, diagonal	5	0.23104686(3)1416	0.20791816(2),1417 0.2085(2)1418
(4+1)-d hypercubic, nn	8	0.1461593(2),1419 0.1461582(3)1420	0.1288557(5)1421
(4+1)-d bcc	16	0.075582(17),1422 0.0755850(3),1423 0.07558515(1)1424	0.063763395(5)1425
(5+1)-d hypercubic, diagonal	6	0.18651358(2)1426	0.170615155(5),1427 0.1714(1) 1428
(5+1)-d hypercubic, nn	10	0.1123373(2)1429	0.1016796(5)1430
(5+1)-d hypercubic bcc	32	0.035967(23),1431 0.035972540(3)1432	0.0314566318(5)1433
(6+1)-d hypercubic, diagonal	7	0.15654718(1)1434	0.145089946(3),1435 0.14581436
(6+1)-d hypercubic, nn	12	0.0913087(2)1437	0.0841997(14)1438
(6+1)-d hypercubic bcc	64	0.017333051(2)1439	0.01565938296(10)1440
(7+1)-d hypercubic, diagonal	8	0.135004176(10)1441	0.126387509(3),1442 0.1270(1) 1443
(7+1)-d hypercubic,nn	14	0.07699336(7)1444	0.07195(5)1445
(7+1)-d bcc	128	0.008 432 989(2)1446	0.007 818 371 82(6)1447

nn = nearest neighbors. For a (d + 1)-dimensional hypercubic system, the hypercube is in d dimensions and the time direction points to the 2D nearest neighbors.

Directed percolation with multiple neighbors

(1+1)-d square with z NN, square lattice for z odd, tilted square lattice for z even

Lattice	z	Site percolation threshold	Bond percolation threshold
(1+1)-d square	3		0.4395(3),1448
(1+1)-d square	5		0.2249(3)1449
(1+1)-d square	7		0.1470(2)1450
(1+1)-d square	9		0.1081(2)1451
(1+1)-d square	11		0.0851(2)1452
(1+1)-d square	13		0.0701(2)1453
(1+1)-d tilted sq	2		0.6447(2)1454
(1+1)-d tilted sq	4		0.3272(2)1455
(1+1)-d tilted sq	6		0.2121(3)1456
(1+1)-d tilted sq	8		0.1553(3)1457
(1+1)-d tilted sq	10		0.1220(2)1458
(1+1)-d tilted sq	12		0.0999(2)1459

For large z, pc ~ 1/z ¹⁴⁶⁰

Site-Bond Directed Percolation

p_b = bond threshold

p_s = site threshold

Site-bond percolation is equivalent to having different probabilities of connections:

P_0 = probability that no sites are connected

P_2 = probability that exactly one descendant is connected to the upper vertex (two connected together)

P_3 = probability that both descendants are connected to the original vertex (all three connected together)

Formulas:

P_0 = (1-p_s) + p_s(1-p_b)^2

P_2 = p_s p_b (1-p_b)

P_3 = p_s p_b^2

P_0 + 2P_2 + P_3 = 1

Lattice	z	p_s	p_b	P_0	P_2	P_3
(1+1)-d square 1461	3	0.644701	1	0.126237	0.229062	0.415639
		0.7	0.93585	0.148376	0.196529	0.458567
		0.75	0.88565	0.169703	0.166059	0.498178
		0.8	0.84135	0.192304	0.134616	0.538464
		0.85	0.80190	0.216143	0.102242	0.579373
		0.9	0.76645	0.241215	0.068981	0.620825
		0.95	0.73450	0.267336	0.034889	0.662886
		1	0.705489	0.294511	0	0.705489

Exact critical manifolds of inhomogeneous systems

Inhomogeneous triangular lattice bond percolation¹⁴⁶²

1 − p 1 − p 2 − p 3 + p 1 p 2 p 3 = 0 {\displaystyle 1-p_{1}-p_{2}-p_{3}+p_{1}p_{2}p_{3}=0}

Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation¹⁴⁶³

1 − p 1 p 2 − p 1 p 3 − p 2 p 3 + p 1 p 2 p 3 = 0 {\displaystyle 1-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}+p_{1}p_{2}p_{3}=0}

Inhomogeneous (3,12^2) lattice, site percolation^14641465

1 − 3 ( s 1 s 2 ) 2 + ( s 1 s 2 ) 3 = 0 , {\displaystyle 1-3(s_{1}s_{2})^{2}+(s_{1}s_{2})^{3}=0,} or s 1 s 2 = 1 − 2 sin ⁡ ( π / 18 ) {\displaystyle s_{1}s_{2}=1-2\sin(\pi /18)}

Inhomogeneous union-jack lattice, site percolation with probabilities p 1 , p 2 , p 3 , p 4 {\displaystyle p_{1},p_{2},p_{3},p_{4}} ¹⁴⁶⁶

p 3 = 1 − p 1 ; p 4 = 1 − p 2 {\displaystyle p_{3}=1-p_{1};\qquad p_{4}=1-p_{2}}

Inhomogeneous martini lattice, bond percolation^14671468

1 − ( p 1 p 2 r 3 + p 2 p 3 r 1 + p 1 p 3 r 2 ) − ( p 1 p 2 r 1 r 2 + p 1 p 3 r 1 r 3 + p 2 p 3 r 2 r 3 ) + p 1 p 2 p 3 ( r 1 r 2 + r 1 r 3 + r 2 r 3 ) + r 1 r 2 r 3 ( p 1 p 2 + p 1 p 3 + p 2 p 3 ) − 2 p 1 p 2 p 3 r 1 r 2 r 3 = 0 {\displaystyle 1-(p_{1}p_{2}r_{3}+p_{2}p_{3}r_{1}+p_{1}p_{3}r_{2})-(p_{1}p_{2}r_{1}r_{2}+p_{1}p_{3}r_{1}r_{3}+p_{2}p_{3}r_{2}r_{3})+p_{1}p_{2}p_{3}(r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3})+r_{1}r_{2}r_{3}(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})-2p_{1}p_{2}p_{3}r_{1}r_{2}r_{3}=0}

Inhomogeneous martini lattice, site percolation. r = site in the star

1 − r ( p 1 p 2 + p 1 p 3 + p 2 p 3 − p 1 p 2 p 3 ) = 0 {\displaystyle 1-r(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3}-p_{1}p_{2}p_{3})=0}

Inhomogeneous martini-A (3–7) lattice, bond percolation. Left side (top of "A" to bottom): r 2 ,   p 1 {\displaystyle r_{2},\ p_{1}} . Right side: r 1 ,   p 2 {\displaystyle r_{1},\ p_{2}} . Cross bond:   r 3 {\displaystyle \ r_{3}} .

1 − p 1 r 2 − p 2 r 1 − p 1 p 2 r 3 − p 1 r 1 r 3 − p 2 r 2 r 3 + p 1 p 2 r 1 r 3 + p 1 p 2 r 2 r 3 + p 1 r 1 r 2 r 3 + p 2 r 1 r 2 r 3 − p 1 p 2 r 1 r 2 r 3 = 0 {\displaystyle 1-p_{1}r_{2}-p_{2}r_{1}-p_{1}p_{2}r_{3}-p_{1}r_{1}r_{3}-p_{2}r_{2}r_{3}+p_{1}p_{2}r_{1}r_{3}+p_{1}p_{2}r_{2}r_{3}+p_{1}r_{1}r_{2}r_{3}+p_{2}r_{1}r_{2}r_{3}-p_{1}p_{2}r_{1}r_{2}r_{3}=0}

Inhomogeneous martini-B (3–5) lattice, bond percolation

Inhomogeneous martini lattice with outside enclosing triangle of bonds, probabilities y , x , z {\displaystyle y,x,z} from inside to outside, bond percolation¹⁴⁶⁹

1 − 3 z + z 3 − ( 1 − z 2 ) [ 3 x 2 y ( 1 + y − y 2 ) ( 1 + z ) + x 3 y 2 ( 3 − 2 y ) ( 1 + 2 z ) ] = 0 {\displaystyle 1-3z+z^{3}-(1-z^{2})[3x^{2}y(1+y-y^{2})(1+z)+x^{3}y^{2}(3-2y)(1+2z)]=0}

Inhomogeneous checkerboard lattice, bond percolation^14701471

1 − ( p 1 p 2 + p 1 p 3 + p 1 p 4 + p 2 p 3 + p 2 p 4 + p 3 p 4 ) + p 1 p 2 p 3 + p 1 p 2 p 4 + p 1 p 3 p 4 + p 2 p 3 p 4 = 0 {\displaystyle 1-(p_{1}p_{2}+p_{1}p_{3}+p_{1}p_{4}+p_{2}p_{3}+p_{2}p_{4}+p_{3}p_{4})+p_{1}p_{2}p_{3}+p_{1}p_{2}p_{4}+p_{1}p_{3}p_{4}+p_{2}p_{3}p_{4}=0}

Inhomogeneous bow-tie lattice, bond percolation^14721473

1 − ( p 1 p 2 + p 1 p 3 + p 1 p 4 + p 2 p 3 + p 2 p 4 + p 3 p 4 ) + p 1 p 2 p 3 + p 1 p 2 p 4 + p 1 p 3 p 4 + p 2 p 3 p 4 − u ( 1 − p 1 p 2 − p 3 p 4 + p 1 p 2 p 3 p 4 ) = 0 {\displaystyle 1-(p_{1}p_{2}+p_{1}p_{3}+p_{1}p_{4}+p_{2}p_{3}+p_{2}p_{4}+p_{3}p_{4})+p_{1}p_{2}p_{3}+p_{1}p_{2}p_{4}+p_{1}p_{3}p_{4}+p_{2}p_{3}p_{4}-u(1-p_{1}p_{2}-p_{3}p_{4}+p_{1}p_{2}p_{3}p_{4})=0}

where p 1 , p 2 , p 3 , p 4 {\displaystyle p_{1},p_{2},p_{3},p_{4}} are the four bonds around the square and u {\displaystyle u} is the diagonal bond connecting the vertex between bonds p 4 , p 1 {\displaystyle p_{4},p_{1}} and p 2 , p 3 {\displaystyle p_{2},p_{3}} .

See also

• 2D percolation cluster
• Bootstrap percolation
• Directed percolation
• Effective medium approximations
• Epidemic models on lattices
• Graph theory
• Network science
• Percolation
• Percolation critical exponents
• Percolation theory
• Continuum percolation theory
• Random sequential adsorption
• Uniform tilings

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375. Ding, Chengxiang; Zhe Fu. Wenan Guo; F. Y. Wu (2010). "Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis". Physical Review E. 81 (6): 061111. arXiv:1001.1488. Bibcode:2010PhRvE..81f1111D. doi:10.1103/PhysRevE.81.061111. PMID 20866382. S2CID 29625353. /wiki/ArXiv_(identifier) ↩

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377. Haji Akbari, Amir; R. M. Ziff (2009). "Percolation in networks with voids and bottlenecks". Physical Review E. 79 (2): 021118. arXiv:0811.4575. Bibcode:2009PhRvE..79b1118H. doi:10.1103/PhysRevE.79.021118. PMID 19391717. S2CID 2554311. /wiki/ArXiv_(identifier) ↩

378. Haji Akbari, Amir; R. M. Ziff (2009). "Percolation in networks with voids and bottlenecks". Physical Review E. 79 (2): 021118. arXiv:0811.4575. Bibcode:2009PhRvE..79b1118H. doi:10.1103/PhysRevE.79.021118. PMID 19391717. S2CID 2554311. /wiki/ArXiv_(identifier) ↩

379. Haji Akbari, Amir; R. M. Ziff (2009). "Percolation in networks with voids and bottlenecks". Physical Review E. 79 (2): 021118. arXiv:0811.4575. Bibcode:2009PhRvE..79b1118H. doi:10.1103/PhysRevE.79.021118. PMID 19391717. S2CID 2554311. /wiki/ArXiv_(identifier) ↩

380. Haji Akbari, Amir; R. M. Ziff (2009). "Percolation in networks with voids and bottlenecks". Physical Review E. 79 (2): 021118. arXiv:0811.4575. Bibcode:2009PhRvE..79b1118H. doi:10.1103/PhysRevE.79.021118. PMID 19391717. S2CID 2554311. /wiki/ArXiv_(identifier) ↩

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384. Lebrecht, W.; P. M. Centres; A. J. Ramirez-Pastor (2019). "Analytical approximation of the site percolation thresholds for monomers and dimers on two-dimensional lattices". Physica A. 516: 133–143. Bibcode:2019PhyA..516..133L. doi:10.1016/j.physa.2018.10.023. S2CID 125418069. /wiki/Bibcode_(identifier) ↩

385. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

386. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

387. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

388. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

389. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

390. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

391. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

392. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

393. Longone, Pablo; P.M. Centres; A. J. Ramirez-Pastor (2019). "Percolation of aligned rigid rods on two-dimensional triangular lattices". Physical Review E. 100 (5): 052104. arXiv:1906.03966. Bibcode:2019PhRvE.100e2104L. doi:10.1103/PhysRevE.100.052104. PMID 31870027. S2CID 182953009. /wiki/ArXiv_(identifier) ↩

394. Budinski-Petkovic, Lj; I. Loncarevic; Z. M. Jacsik; S. B. Vrhovac (2016). "Jamming and percolation in random sequential adsorption of extended objects on a triangular lattice with quenched impurities". Journal of Statistical Mechanics: Theory and Experiment. 2016 (5): 053101. Bibcode:2016JSMTE..05.3101B. doi:10.1088/1742-5468/2016/05/053101. S2CID 3913989. /wiki/Bibcode_(identifier) ↩

395. Cherkasova, V. A.; Yu. Yu. Tarasevich; N. I. Lebovka; N.V. Vygornitskii (2010). "Percolation of the aligned dimers on a square lattice". Eur. Phys. J. B. 74 (2): 205–209. arXiv:0912.0778. Bibcode:2010EPJB...74..205C. doi:10.1140/epjb/e2010-00089-2. S2CID 118485353. /wiki/ArXiv_(identifier) ↩

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399. Lebrecht, W.; P. M. Centres; A. J. Ramirez-Pastor (2019). "Analytical approximation of the site percolation thresholds for monomers and dimers on two-dimensional lattices". Physica A. 516: 133–143. Bibcode:2019PhyA..516..133L. doi:10.1016/j.physa.2018.10.023. S2CID 125418069. /wiki/Bibcode_(identifier) ↩

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405. Vanderwalle, N.; S. Galam; M. Kramer (2000). "A new universality for random sequential deposition of needles". Eur. Phys. J. B. 14 (3): 407–410. arXiv:cond-mat/0004271. Bibcode:2000EPJB...14..407V. doi:10.1007/s100510051047. S2CID 11142384. /wiki/ArXiv_(identifier) ↩

406. Vanderwalle, N.; S. Galam; M. Kramer (2000). "A new universality for random sequential deposition of needles". Eur. Phys. J. B. 14 (3): 407–410. arXiv:cond-mat/0004271. Bibcode:2000EPJB...14..407V. doi:10.1007/s100510051047. S2CID 11142384. /wiki/ArXiv_(identifier) ↩

407. Vanderwalle, N.; S. Galam; M. Kramer (2000). "A new universality for random sequential deposition of needles". Eur. Phys. J. B. 14 (3): 407–410. arXiv:cond-mat/0004271. Bibcode:2000EPJB...14..407V. doi:10.1007/s100510051047. S2CID 11142384. /wiki/ArXiv_(identifier) ↩

408. Leroyer, Y.; E. Pommiers (1994). "Monte Carlo analysis of percolation of line segments on a square lattice". Physical Review B. 50 (5): 2795–2799. arXiv:cond-mat/9312066. Bibcode:1994PhRvB..50.2795L. doi:10.1103/PhysRevB.50.2795. PMID 9976520. S2CID 119495907. /wiki/ArXiv_(identifier) ↩

409. Vanderwalle, N.; S. Galam; M. Kramer (2000). "A new universality for random sequential deposition of needles". Eur. Phys. J. B. 14 (3): 407–410. arXiv:cond-mat/0004271. Bibcode:2000EPJB...14..407V. doi:10.1007/s100510051047. S2CID 11142384. /wiki/ArXiv_(identifier) ↩

410. Leroyer, Y.; E. Pommiers (1994). "Monte Carlo analysis of percolation of line segments on a square lattice". Physical Review B. 50 (5): 2795–2799. arXiv:cond-mat/9312066. Bibcode:1994PhRvB..50.2795L. doi:10.1103/PhysRevB.50.2795. PMID 9976520. S2CID 119495907. /wiki/ArXiv_(identifier) ↩

411. Leroyer, Y.; E. Pommiers (1994). "Monte Carlo analysis of percolation of line segments on a square lattice". Physical Review B. 50 (5): 2795–2799. arXiv:cond-mat/9312066. Bibcode:1994PhRvB..50.2795L. doi:10.1103/PhysRevB.50.2795. PMID 9976520. S2CID 119495907. /wiki/ArXiv_(identifier) ↩

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415. Haji-Akbari, A.; Nasim Haji-Akbari; Robert M. Ziff (2015). "Dimer Covering and Percolation Frustration". Physical Review E. 92 (3): 032134. arXiv:1507.04411. Bibcode:2015PhRvE..92c2134H. doi:10.1103/PhysRevE.92.032134. PMID 26465453. S2CID 34100812. /wiki/ArXiv_(identifier) ↩

416. Haji-Akbari, A.; Nasim Haji-Akbari; Robert M. Ziff (2015). "Dimer Covering and Percolation Frustration". Physical Review E. 92 (3): 032134. arXiv:1507.04411. Bibcode:2015PhRvE..92c2134H. doi:10.1103/PhysRevE.92.032134. PMID 26465453. S2CID 34100812. /wiki/ArXiv_(identifier) ↩

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426. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

427. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

428. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

429. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

430. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

431. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

432. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

433. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

434. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

435. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

436. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

437. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

438. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

439. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

440. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

441. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

442. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

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444. Cornette, V.; A.J. Ramirez-Pastor; F. Nieto (2003). "Percolation of polyatomic species on a square lattice". European Physical Journal B. 36 (3): 391–399. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. S2CID 119852589. /wiki/Bibcode_(identifier) ↩

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488. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

489. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

490. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

491. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

492. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

493. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

494. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

495. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

496. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

497. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

498. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

499. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

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501. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

502. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

503. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proceedings of the Royal Society A. 460 (5): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

504. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

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506. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

507. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

508. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

509. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

510. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proceedings of the Royal Society A. 460 (5): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

511. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

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513. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proceedings of the Royal Society A. 460 (5): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

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515. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

516. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proceedings of the Royal Society A. 460 (5): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

517. Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A. 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. /wiki/Bibcode_(identifier) ↩

518. Li, Jiantong; Mikael Östling (2016). "Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses". Physica A. 462: 940–950. Bibcode:2016PhyA..462..940L. doi:10.1016/j.physa.2016.06.020. http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01 ↩

519. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proceedings of the Royal Society A. 460 (5): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

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537. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

538. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

539. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

540. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

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549. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

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551. Mertens, Stephan; Cristopher Moore (2012). "Continuum percolation thresholds in two dimensions". Physical Review E. 86 (6): 061109. arXiv:1209.4936. Bibcode:2012PhRvE..86f1109M. doi:10.1103/PhysRevE.86.061109. PMID 23367895. S2CID 15107275. /wiki/ArXiv_(identifier) ↩

552. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

553. Baker, Don R.; Gerald Paul; Sameet Sreenivasan; H. Eugene Stanley (2002). "Continuum percolation threshold for interpenetrating squares and cubes". Physical Review E. 66 (4): 046136 [5 pages]. arXiv:cond-mat/0203235. Bibcode:2002PhRvE..66d6136B. doi:10.1103/PhysRevE.66.046136. PMID 12443288. S2CID 9561586. /wiki/ArXiv_(identifier) ↩

554. Mertens, Stephan; Cristopher Moore (2012). "Continuum percolation thresholds in two dimensions". Physical Review E. 86 (6): 061109. arXiv:1209.4936. Bibcode:2012PhRvE..86f1109M. doi:10.1103/PhysRevE.86.061109. PMID 23367895. S2CID 15107275. /wiki/ArXiv_(identifier) ↩

555. Baker, Don R.; Gerald Paul; Sameet Sreenivasan; H. Eugene Stanley (2002). "Continuum percolation threshold for interpenetrating squares and cubes". Physical Review E. 66 (4): 046136 [5 pages]. arXiv:cond-mat/0203235. Bibcode:2002PhRvE..66d6136B. doi:10.1103/PhysRevE.66.046136. PMID 12443288. S2CID 9561586. /wiki/ArXiv_(identifier) ↩

556. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

557. Mertens, Stephan; Cristopher Moore (2012). "Continuum percolation thresholds in two dimensions". Physical Review E. 86 (6): 061109. arXiv:1209.4936. Bibcode:2012PhRvE..86f1109M. doi:10.1103/PhysRevE.86.061109. PMID 23367895. S2CID 15107275. /wiki/ArXiv_(identifier) ↩

558. Baker, Don R.; Gerald Paul; Sameet Sreenivasan; H. Eugene Stanley (2002). "Continuum percolation threshold for interpenetrating squares and cubes". Physical Review E. 66 (4): 046136 [5 pages]. arXiv:cond-mat/0203235. Bibcode:2002PhRvE..66d6136B. doi:10.1103/PhysRevE.66.046136. PMID 12443288. S2CID 9561586. /wiki/ArXiv_(identifier) ↩

559. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

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562. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

563. Baker, Don R.; Gerald Paul; Sameet Sreenivasan; H. Eugene Stanley (2002). "Continuum percolation threshold for interpenetrating squares and cubes". Physical Review E. 66 (4): 046136 [5 pages]. arXiv:cond-mat/0203235. Bibcode:2002PhRvE..66d6136B. doi:10.1103/PhysRevE.66.046136. PMID 12443288. S2CID 9561586. /wiki/ArXiv_(identifier) ↩

564. Baker, Don R.; Gerald Paul; Sameet Sreenivasan; H. Eugene Stanley (2002). "Continuum percolation threshold for interpenetrating squares and cubes". Physical Review E. 66 (4): 046136 [5 pages]. arXiv:cond-mat/0203235. Bibcode:2002PhRvE..66d6136B. doi:10.1103/PhysRevE.66.046136. PMID 12443288. S2CID 9561586. /wiki/ArXiv_(identifier) ↩

565. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

566. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

567. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

568. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

569. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

570. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

571. Lin, Jianjun; Chen, Huisu (2019). "Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses". Powder Technology. 347: 17–26. doi:10.1016/j.powtec.2019.02.036. S2CID 104332397. /wiki/Doi_(identifier) ↩

572. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

573. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

574. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

575. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

576. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

577. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

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580. Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E. 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. PMID 23944408. S2CID 21438506. https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 ↩

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827. Kurzawski, Ł.; K. Malarz (2012). "Simple cubic random-site percolation thresholds for complex neighbourhoods". Rep. Math. Phys. 70 (2): 163–169. arXiv:1111.3254. Bibcode:2012RpMP...70..163K. CiteSeerX 10.1.1.743.1726. doi:10.1016/S0034-4877(12)60036-6. S2CID 119120046. /wiki/ArXiv_(identifier) ↩

828. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

829. Xun, Zhipeng; Robert M. Ziff (2020). "Bond percolation on simple cubic lattices with extended neighborhoods". Physical Review E. 102 (4): 012102. arXiv:2001.00349. Bibcode:2020PhRvE.102a2102X. doi:10.1103/PhysRevE.102.012102. PMID 32795057. S2CID 209531616. /wiki/ArXiv_(identifier) ↩

830. Dalton, N. W.; C. Domb; M. F. Sykes (1964). "Dependence of critical concentration of a dilute ferromagnet on the range of interaction". Proc. Phys. Soc. 83 (3): 496–498. doi:10.1088/0370-1328/83/3/118. /wiki/Doi_(identifier) ↩

831. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

832. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

833. Harter, T. (2005). "Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields". Physical Review E. 72 (2): 026120. Bibcode:2005PhRvE..72b6120H. doi:10.1103/PhysRevE.72.026120. PMID 16196657. S2CID 2708506. /wiki/Bibcode_(identifier) ↩

834. Malarz, Krzysztof (2015). "Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors". Physical Review E. 91 (4): 043301. arXiv:1501.01586. Bibcode:2015PhRvE..91d3301M. doi:10.1103/PhysRevE.91.043301. PMID 25974606. S2CID 37943657. /wiki/ArXiv_(identifier) ↩

835. Xun, Zhipeng; Robert M. Ziff (2020). "Bond percolation on simple cubic lattices with extended neighborhoods". Physical Review E. 102 (4): 012102. arXiv:2001.00349. Bibcode:2020PhRvE.102a2102X. doi:10.1103/PhysRevE.102.012102. PMID 32795057. S2CID 209531616. /wiki/ArXiv_(identifier) ↩

836. Kurzawski, Ł.; K. Malarz (2012). "Simple cubic random-site percolation thresholds for complex neighbourhoods". Rep. Math. Phys. 70 (2): 163–169. arXiv:1111.3254. Bibcode:2012RpMP...70..163K. CiteSeerX 10.1.1.743.1726. doi:10.1016/S0034-4877(12)60036-6. S2CID 119120046. /wiki/ArXiv_(identifier) ↩

837. Xun, Zhipeng; Robert M. Ziff (2020). "Bond percolation on simple cubic lattices with extended neighborhoods". Physical Review E. 102 (4): 012102. arXiv:2001.00349. Bibcode:2020PhRvE.102a2102X. doi:10.1103/PhysRevE.102.012102. PMID 32795057. S2CID 209531616. /wiki/ArXiv_(identifier) ↩

838. Malarz, Krzysztof (2015). "Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors". Physical Review E. 91 (4): 043301. arXiv:1501.01586. Bibcode:2015PhRvE..91d3301M. doi:10.1103/PhysRevE.91.043301. PMID 25974606. S2CID 37943657. /wiki/ArXiv_(identifier) ↩

839. Xun, Zhipeng; Robert M. Ziff (2020). "Bond percolation on simple cubic lattices with extended neighborhoods". Physical Review E. 102 (4): 012102. arXiv:2001.00349. Bibcode:2020PhRvE.102a2102X. doi:10.1103/PhysRevE.102.012102. PMID 32795057. S2CID 209531616. /wiki/ArXiv_(identifier) ↩

840. Malarz, Krzysztof (2015). "Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors". Physical Review E. 91 (4): 043301. arXiv:1501.01586. Bibcode:2015PhRvE..91d3301M. doi:10.1103/PhysRevE.91.043301. PMID 25974606. S2CID 37943657. /wiki/ArXiv_(identifier) ↩

841. Dalton, N. W.; C. Domb; M. F. Sykes (1964). "Dependence of critical concentration of a dilute ferromagnet on the range of interaction". Proc. Phys. Soc. 83 (3): 496–498. doi:10.1088/0370-1328/83/3/118. /wiki/Doi_(identifier) ↩

842. Kurzawski, Ł.; K. Malarz (2012). "Simple cubic random-site percolation thresholds for complex neighbourhoods". Rep. Math. Phys. 70 (2): 163–169. arXiv:1111.3254. Bibcode:2012RpMP...70..163K. CiteSeerX 10.1.1.743.1726. doi:10.1016/S0034-4877(12)60036-6. S2CID 119120046. /wiki/ArXiv_(identifier) ↩

843. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

844. Xun, Zhipeng; Robert M. Ziff (2020). "Bond percolation on simple cubic lattices with extended neighborhoods". Physical Review E. 102 (4): 012102. arXiv:2001.00349. Bibcode:2020PhRvE.102a2102X. doi:10.1103/PhysRevE.102.012102. PMID 32795057. S2CID 209531616. /wiki/ArXiv_(identifier) ↩

845. Domb, C.; N. W. Dalton (1966). "Crystal statistics with long-range forces I. The equivalent neighbour model". Proc. Phys. Soc. 89 (4): 859–871. Bibcode:1966PPS....89..859D. doi:10.1088/0370-1328/89/4/311. /wiki/Bibcode_(identifier) ↩

846. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

847. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

848. Malarz, Krzysztof (2015). "Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors". Physical Review E. 91 (4): 043301. arXiv:1501.01586. Bibcode:2015PhRvE..91d3301M. doi:10.1103/PhysRevE.91.043301. PMID 25974606. S2CID 37943657. /wiki/ArXiv_(identifier) ↩

849. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

850. Xun, Zhipeng; Robert M. Ziff (2020). "Bond percolation on simple cubic lattices with extended neighborhoods". Physical Review E. 102 (4): 012102. arXiv:2001.00349. Bibcode:2020PhRvE.102a2102X. doi:10.1103/PhysRevE.102.012102. PMID 32795057. S2CID 209531616. /wiki/ArXiv_(identifier) ↩

851. Domb, C.; N. W. Dalton (1966). "Crystal statistics with long-range forces I. The equivalent neighbour model". Proc. Phys. Soc. 89 (4): 859–871. Bibcode:1966PPS....89..859D. doi:10.1088/0370-1328/89/4/311. /wiki/Bibcode_(identifier) ↩

852. Gawron, T. R.; Marek Cieplak (1991). "Site percolation thresholds of the FCC lattice" (PDF). Acta Physica Polonica A. 80 (3): 461. Bibcode:1991AcPPA..80..461G. doi:10.12693/APhysPolA.80.461. http://przyrbwn.icm.edu.pl/APP/PDF/80/a080z3p38.pdf ↩

853. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

854. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

855. Gawron, T. R.; Marek Cieplak (1991). "Site percolation thresholds of the FCC lattice" (PDF). Acta Physica Polonica A. 80 (3): 461. Bibcode:1991AcPPA..80..461G. doi:10.12693/APhysPolA.80.461. http://przyrbwn.icm.edu.pl/APP/PDF/80/a080z3p38.pdf ↩

856. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

857. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

858. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

859. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

860. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

861. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

862. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

863. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

864. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

865. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

866. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

867. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

868. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

869. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

870. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

871. Xun, Zhipeng; DaPeng Hao; Robert M. Ziff (2022). "Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions". Physical Review E. 105 (2): 024105. arXiv:2111.10975. Bibcode:2022PhRvE.105b4105X. doi:10.1103/PhysRevE.105.024105. PMID 35291074. S2CID 244478657. /wiki/ArXiv_(identifier) ↩

872. Koza, Zbigniew; Kondrat, Grzegorz; Suszczyński, Karol (2014). "Percolation of overlapping squares or cubes on a lattice". Journal of Statistical Mechanics: Theory and Experiment. 2014 (11): P11005. arXiv:1606.07969. Bibcode:2014JSMTE..11..005K. doi:10.1088/1742-5468/2014/11/P11005. S2CID 118623466. /wiki/ArXiv_(identifier) ↩

873. Koza, Zbigniew; Kondrat, Grzegorz; Suszczyński, Karol (2014). "Percolation of overlapping squares or cubes on a lattice". Journal of Statistical Mechanics: Theory and Experiment. 2014 (11): P11005. arXiv:1606.07969. Bibcode:2014JSMTE..11..005K. doi:10.1088/1742-5468/2014/11/P11005. S2CID 118623466. /wiki/ArXiv_(identifier) ↩

874. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

875. Koza, Zbigniew; Kondrat, Grzegorz; Suszczyński, Karol (2014). "Percolation of overlapping squares or cubes on a lattice". Journal of Statistical Mechanics: Theory and Experiment. 2014 (11): P11005. arXiv:1606.07969. Bibcode:2014JSMTE..11..005K. doi:10.1088/1742-5468/2014/11/P11005. S2CID 118623466. /wiki/ArXiv_(identifier) ↩

876. Koza, Zbigniew; Kondrat, Grzegorz; Suszczyński, Karol (2014). "Percolation of overlapping squares or cubes on a lattice". Journal of Statistical Mechanics: Theory and Experiment. 2014 (11): P11005. arXiv:1606.07969. Bibcode:2014JSMTE..11..005K. doi:10.1088/1742-5468/2014/11/P11005. S2CID 118623466. /wiki/ArXiv_(identifier) ↩

877. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

878. Koza, Zbigniew; Kondrat, Grzegorz; Suszczyński, Karol (2014). "Percolation of overlapping squares or cubes on a lattice". Journal of Statistical Mechanics: Theory and Experiment. 2014 (11): P11005. arXiv:1606.07969. Bibcode:2014JSMTE..11..005K. doi:10.1088/1742-5468/2014/11/P11005. S2CID 118623466. /wiki/ArXiv_(identifier) ↩

879. Koza, Zbigniew; Kondrat, Grzegorz; Suszczyński, Karol (2014). "Percolation of overlapping squares or cubes on a lattice". Journal of Statistical Mechanics: Theory and Experiment. 2014 (11): P11005. arXiv:1606.07969. Bibcode:2014JSMTE..11..005K. doi:10.1088/1742-5468/2014/11/P11005. S2CID 118623466. /wiki/ArXiv_(identifier) ↩

880. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

881. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

882. Sykes, M. F.; Rehr, J. J.; Glen, Maureen (1996). "A note on the percolation probabilities of pairs of closely similar lattices". Mathematical Proceedings of the Cambridge Philosophical Society. 76: 389–392. doi:10.1017/S0305004100049021. S2CID 96528423. /wiki/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society ↩

883. Weber, H.; W. Paul (1996). "Penetrant diffusion in frozen polymer matrices: A finite-size scaling study of free volume percolation". Physical Review E. 54 (4): 3999–4007. Bibcode:1996PhRvE..54.3999W. doi:10.1103/PhysRevE.54.3999. PMID 9965547. /wiki/Bibcode_(identifier) ↩

884. Mitra, S.; D. Saha; A. Sensharma (2022). "Percolation in a simple cubic lattice with distortion". Physical Review E. 106 (3): 034109. arXiv:2207.12079. Bibcode:2022PhRvE.106c4109M. doi:10.1103/PhysRevE.106.034109. PMID 36266842. /wiki/ArXiv_(identifier) ↩

885. Mitra, S.; D. Saha; A. Sensharma (2022). "Percolation in a simple cubic lattice with distortion". Physical Review E. 106 (3): 034109. arXiv:2207.12079. Bibcode:2022PhRvE.106c4109M. doi:10.1103/PhysRevE.106.034109. PMID 36266842. /wiki/ArXiv_(identifier) ↩

886. Mitra, S.; D. Saha; A. Sensharma (2022). "Percolation in a simple cubic lattice with distortion". Physical Review E. 106 (3): 034109. arXiv:2207.12079. Bibcode:2022PhRvE.106c4109M. doi:10.1103/PhysRevE.106.034109. PMID 36266842. /wiki/ArXiv_(identifier) ↩

887. Mitra, S.; D. Saha; A. Sensharma (2022). "Percolation in a simple cubic lattice with distortion". Physical Review E. 106 (3): 034109. arXiv:2207.12079. Bibcode:2022PhRvE.106c4109M. doi:10.1103/PhysRevE.106.034109. PMID 36266842. /wiki/ArXiv_(identifier) ↩

888. Mitra, S.; D. Saha; A. Sensharma (2022). "Percolation in a simple cubic lattice with distortion". Physical Review E. 106 (3): 034109. arXiv:2207.12079. Bibcode:2022PhRvE.106c4109M. doi:10.1103/PhysRevE.106.034109. PMID 36266842. /wiki/ArXiv_(identifier) ↩

889. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

890. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

891. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

892. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

893. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

894. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

895. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

896. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

897. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

898. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

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901. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

902. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

903. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

904. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

905. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

906. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

907. Mecke, K. R.; Seyfried, A (2002). "Strong dependence of percolation thresholds on polydispersity". Europhysics Letters (EPL). 58 (1): 28–34. Bibcode:2002EL.....58...28M. doi:10.1209/epl/i2002-00601-y. S2CID 250737562. /wiki/Bibcode_(identifier) ↩

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917. Lorenz, C. D.; R. M. Ziff (2000). "Precise determination of the critical percolation threshold for the three dimensional Swiss cheese model using a growth algorithm" (PDF). J. Chem. Phys. 114 (8): 3659. Bibcode:2001JChPh.114.3659L. doi:10.1063/1.1338506. hdl:2027.42/70114. https://deepblue.lib.umich.edu/bitstream/2027.42/70114/2/JCPSA6-114-8-3659-1.pdf ↩

918. Lin, Jianjun; Chen, Huisu; Xu, Wenxiang (2018). "Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems". Physical Review E. 98 (1): 012134. Bibcode:2018PhRvE..98a2134L. doi:10.1103/PhysRevE.98.012134. PMID 30110832. S2CID 52017287. /wiki/Bibcode_(identifier) ↩

919. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

920. Li, Mingqi; Chen, Huisu; Lin, Jianjun (January 2020). "Efficient measurement of the percolation threshold for random systems of congruent overlapping ovoids". Powder Technology. 360: 598–607. doi:10.1016/j.powtec.2019.10.044. ISSN 0032-5910. S2CID 208693526. /wiki/Doi_(identifier) ↩

921. Li, Mingqi; Chen, Huisu; Lin, Jianjun (April 2020). "Numerical study for the percolation threshold and transport properties of porous composites comprising non-centrosymmetrical superovoidal pores". Computer Methods in Applied Mechanics and Engineering. 361: 112815. Bibcode:2020CMAME.361k2815L. doi:10.1016/j.cma.2019.112815. ISSN 0045-7825. S2CID 213152892. /wiki/Bibcode_(identifier) ↩

922. Brzeski, Piotr; Kondrat, Grzegorz (2022). "Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous". Journal of Statistical Mechanics: Theory and Experiment. 2022 (5): 053202. doi:10.1088/1742-5468/ac6519. ISSN 1742-5468. https://iopscience.iop.org/article/10.1088/1742-5468/ac6519 ↩

923. Rintoul, M. D.; S. Torquato (1997). "Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model". J. Phys. A: Math. Gen. 30 (16): L585. Bibcode:1997JPhA...30L.585R. CiteSeerX 10.1.1.42.4284. doi:10.1088/0305-4470/30/16/005. /wiki/Bibcode_(identifier) ↩

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925. Lorenz, C. D.; R. M. Ziff (2000). "Precise determination of the critical percolation threshold for the three dimensional Swiss cheese model using a growth algorithm" (PDF). J. Chem. Phys. 114 (8): 3659. Bibcode:2001JChPh.114.3659L. doi:10.1063/1.1338506. hdl:2027.42/70114. https://deepblue.lib.umich.edu/bitstream/2027.42/70114/2/JCPSA6-114-8-3659-1.pdf ↩

926. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

927. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

928. Gori, Giacomo; Andrea Trombettoni (2015). "Conformal invariance in three dimensional percolation". Journal of Statistical Mechanics: Theory and Experiment. 2015 (7): P07014. arXiv:1504.07209. Bibcode:2015JSMTE..07..014G. doi:10.1088/1742-5468/2015/07/P07014. S2CID 119292052. /wiki/ArXiv_(identifier) ↩

929. Balberg, I.; N. Binenbaum (1984). "Percolation thresholds in the three-dimensional sticks system". Physical Review Letters. 52 (17): 1465. Bibcode:1984PhRvL..52.1465B. doi:10.1103/PhysRevLett.52.1465. /wiki/Bibcode_(identifier) ↩

930. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

931. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

932. Lin, Jianjun; Chen, Huisu; Xu, Wenxiang (2018). "Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems". Physical Review E. 98 (1): 012134. Bibcode:2018PhRvE..98a2134L. doi:10.1103/PhysRevE.98.012134. PMID 30110832. S2CID 52017287. /wiki/Bibcode_(identifier) ↩

933. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

934. Li, Mingqi; Chen, Huisu; Lin, Jianjun (January 2020). "Efficient measurement of the percolation threshold for random systems of congruent overlapping ovoids". Powder Technology. 360: 598–607. doi:10.1016/j.powtec.2019.10.044. ISSN 0032-5910. S2CID 208693526. /wiki/Doi_(identifier) ↩

935. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

936. Lin, Jianjun; Chen, Huisu; Xu, Wenxiang (2018). "Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems". Physical Review E. 98 (1): 012134. Bibcode:2018PhRvE..98a2134L. doi:10.1103/PhysRevE.98.012134. PMID 30110832. S2CID 52017287. /wiki/Bibcode_(identifier) ↩

937. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

938. Li, Mingqi; Chen, Huisu; Lin, Jianjun (January 2020). "Efficient measurement of the percolation threshold for random systems of congruent overlapping ovoids". Powder Technology. 360: 598–607. doi:10.1016/j.powtec.2019.10.044. ISSN 0032-5910. S2CID 208693526. /wiki/Doi_(identifier) ↩

939. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

940. Lin, Jianjun; Chen, Huisu; Xu, Wenxiang (2018). "Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems". Physical Review E. 98 (1): 012134. Bibcode:2018PhRvE..98a2134L. doi:10.1103/PhysRevE.98.012134. PMID 30110832. S2CID 52017287. /wiki/Bibcode_(identifier) ↩

941. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

942. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

943. Li, Mingqi; Chen, Huisu; Lin, Jianjun (January 2020). "Efficient measurement of the percolation threshold for random systems of congruent overlapping ovoids". Powder Technology. 360: 598–607. doi:10.1016/j.powtec.2019.10.044. ISSN 0032-5910. S2CID 208693526. /wiki/Doi_(identifier) ↩

944. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

945. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

946. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

947. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

948. Lin, Jianjun; Chen, Huisu; Xu, Wenxiang (2018). "Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems". Physical Review E. 98 (1): 012134. Bibcode:2018PhRvE..98a2134L. doi:10.1103/PhysRevE.98.012134. PMID 30110832. S2CID 52017287. /wiki/Bibcode_(identifier) ↩

949. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

950. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

951. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

952. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

953. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

954. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

955. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

956. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

957. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

958. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

959. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

960. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

961. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

962. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

963. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

964. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

965. Lin, Jianjun; Chen, Huisu; Xu, Wenxiang (2018). "Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems". Physical Review E. 98 (1): 012134. Bibcode:2018PhRvE..98a2134L. doi:10.1103/PhysRevE.98.012134. PMID 30110832. S2CID 52017287. /wiki/Bibcode_(identifier) ↩

966. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

967. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

968. Lin, Jianjun; Chen, Huisu; Xu, Wenxiang (2018). "Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems". Physical Review E. 98 (1): 012134. Bibcode:2018PhRvE..98a2134L. doi:10.1103/PhysRevE.98.012134. PMID 30110832. S2CID 52017287. /wiki/Bibcode_(identifier) ↩

969. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

970. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

971. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

972. Yi, Y.-B.; A. M. Sastry (2004). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proc. R. Soc. Lond. A. 460 (2048): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. S2CID 2475482. /wiki/Bibcode_(identifier) ↩

973. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

974. Garboczi, E. J.; K. A. Snyder; J. F. Douglas (1995). "Geometrical percolation threshold of overlapping ellipsoids". Physical Review E. 52 (1): 819–827. Bibcode:1995PhRvE..52..819G. doi:10.1103/PhysRevE.52.819. PMID 9963485. https://zenodo.org/record/1233793 ↩

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996. Torquato, S.; Y. Jiao (2012). "Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles". Physical Review E. 87 (2): 022111. arXiv:1210.0134. Bibcode:2013PhRvE..87b2111T. doi:10.1103/PhysRevE.87.022111. PMID 23496464. S2CID 11417012. /wiki/ArXiv_(identifier) ↩

997. Torquato, S.; Y. Jiao (2012). "Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles". Physical Review E. 87 (2): 022111. arXiv:1210.0134. Bibcode:2013PhRvE..87b2111T. doi:10.1103/PhysRevE.87.022111. PMID 23496464. S2CID 11417012. /wiki/ArXiv_(identifier) ↩

998. Torquato, S.; Y. Jiao (2012). "Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles". Physical Review E. 87 (2): 022111. arXiv:1210.0134. Bibcode:2013PhRvE..87b2111T. doi:10.1103/PhysRevE.87.022111. PMID 23496464. S2CID 11417012. /wiki/ArXiv_(identifier) ↩

999. Baker, Don R.; Gerald Paul; Sameet Sreenivasan; H. Eugene Stanley (2002). "Continuum percolation threshold for interpenetrating squares and cubes". Physical Review E. 66 (4): 046136 [5 pages]. arXiv:cond-mat/0203235. Bibcode:2002PhRvE..66d6136B. doi:10.1103/PhysRevE.66.046136. PMID 12443288. S2CID 9561586. /wiki/ArXiv_(identifier) ↩

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1002. Torquato, S.; Y. Jiao (2012). "Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles". Physical Review E. 87 (2): 022111. arXiv:1210.0134. Bibcode:2013PhRvE..87b2111T. doi:10.1103/PhysRevE.87.022111. PMID 23496464. S2CID 11417012. /wiki/ArXiv_(identifier) ↩

1003. Torquato, S.; Y. Jiao (2012). "Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles". Physical Review E. 87 (2): 022111. arXiv:1210.0134. Bibcode:2013PhRvE..87b2111T. doi:10.1103/PhysRevE.87.022111. PMID 23496464. S2CID 11417012. /wiki/ArXiv_(identifier) ↩

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1012. Xu, Wenxiang; Xianglong Su; Yang Jiao (2016). "Continuum percolation of congruent overlapping spherocylinders". Physical Review E. 93 (3): 032122. Bibcode:2016PhRvE..94c2122X. doi:10.1103/PhysRevE.94.032122. PMID 27078307. /wiki/Bibcode_(identifier) ↩

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1017. Yi, Y. B.; K. Esmail (2012). "Computational measurement of void percolation thresholds of oblate particles and thin plate composites". J. Appl. Phys. 111 (12): 124903–124903–6. Bibcode:2012JAP...111l4903Y. doi:10.1063/1.4730333. /wiki/Bibcode_(identifier) ↩

1018. Priour, Jr., D. J.; N. J. McGuigan (2018). "Percolation through voids around randomly oriented polyhedra and axially symmetric grains". Physical Review Letters. 121 (22): 225701. arXiv:1801.09970. Bibcode:2018PhRvL.121v5701P. doi:10.1103/PhysRevLett.121.225701. PMID 30547614. S2CID 119185480. /wiki/ArXiv_(identifier) ↩

1019. Novak, Igor L.; Fei Gao; Pavel Kraikivski; Boris M. Slepchenko (2011). "Diffusion amid random overlapping obstacles: Similarities, invariants, approximations". J. Chem. Phys. 134 (15): 154104. Bibcode:2011JChPh.134o4104N. doi:10.1063/1.3578684. PMC 3094463. PMID 21513372. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3094463 ↩

1020. Novak, Igor L.; Fei Gao; Pavel Kraikivski; Boris M. Slepchenko (2011). "Diffusion amid random overlapping obstacles: Similarities, invariants, approximations". J. Chem. Phys. 134 (15): 154104. Bibcode:2011JChPh.134o4104N. doi:10.1063/1.3578684. PMC 3094463. PMID 21513372. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3094463 ↩

1021. Novak, Igor L.; Fei Gao; Pavel Kraikivski; Boris M. Slepchenko (2011). "Diffusion amid random overlapping obstacles: Similarities, invariants, approximations". J. Chem. Phys. 134 (15): 154104. Bibcode:2011JChPh.134o4104N. doi:10.1063/1.3578684. PMC 3094463. PMID 21513372. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3094463 ↩

1022. Novak, Igor L.; Fei Gao; Pavel Kraikivski; Boris M. Slepchenko (2011). "Diffusion amid random overlapping obstacles: Similarities, invariants, approximations". J. Chem. Phys. 134 (15): 154104. Bibcode:2011JChPh.134o4104N. doi:10.1063/1.3578684. PMC 3094463. PMID 21513372. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3094463 ↩

1023. Yi, Y. B.; K. Esmail (2012). "Computational measurement of void percolation thresholds of oblate particles and thin plate composites". J. Appl. Phys. 111 (12): 124903–124903–6. Bibcode:2012JAP...111l4903Y. doi:10.1063/1.4730333. /wiki/Bibcode_(identifier) ↩

1024. Novak, Igor L.; Fei Gao; Pavel Kraikivski; Boris M. Slepchenko (2011). "Diffusion amid random overlapping obstacles: Similarities, invariants, approximations". J. Chem. Phys. 134 (15): 154104. Bibcode:2011JChPh.134o4104N. doi:10.1063/1.3578684. PMC 3094463. PMID 21513372. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3094463 ↩

1025. Novak, Igor L.; Fei Gao; Pavel Kraikivski; Boris M. Slepchenko (2011). "Diffusion amid random overlapping obstacles: Similarities, invariants, approximations". J. Chem. Phys. 134 (15): 154104. Bibcode:2011JChPh.134o4104N. doi:10.1063/1.3578684. PMC 3094463. PMID 21513372. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3094463 ↩

1026. Novak, Igor L.; Fei Gao; Pavel Kraikivski; Boris M. Slepchenko (2011). "Diffusion amid random overlapping obstacles: Similarities, invariants, approximations". J. Chem. Phys. 134 (15): 154104. Bibcode:2011JChPh.134o4104N. doi:10.1063/1.3578684. PMC 3094463. PMID 21513372. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3094463 ↩

1027. Novak, Igor L.; Fei Gao; Pavel Kraikivski; Boris M. Slepchenko (2011). "Diffusion amid random overlapping obstacles: Similarities, invariants, approximations". J. Chem. Phys. 134 (15): 154104. Bibcode:2011JChPh.134o4104N. doi:10.1063/1.3578684. PMC 3094463. PMID 21513372. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3094463 ↩

1028. Yi, Y. B.; K. Esmail (2012). "Computational measurement of void percolation thresholds of oblate particles and thin plate composites". J. Appl. Phys. 111 (12): 124903–124903–6. Bibcode:2012JAP...111l4903Y. doi:10.1063/1.4730333. /wiki/Bibcode_(identifier) ↩

1029. Yi, Y. B.; K. Esmail (2012). "Computational measurement of void percolation thresholds of oblate particles and thin plate composites". J. Appl. Phys. 111 (12): 124903–124903–6. Bibcode:2012JAP...111l4903Y. doi:10.1063/1.4730333. /wiki/Bibcode_(identifier) ↩

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1032. Yi, Y. B. (2006). "Void percolation and conduction of overlapping ellipsoids". Physical Review E. 74 (3): 031112. Bibcode:2006PhRvE..74c1112Y. doi:10.1103/PhysRevE.74.031112. PMID 17025599. /wiki/Bibcode_(identifier) ↩

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1034. Yi, Y. B. (2006). "Void percolation and conduction of overlapping ellipsoids". Physical Review E. 74 (3): 031112. Bibcode:2006PhRvE..74c1112Y. doi:10.1103/PhysRevE.74.031112. PMID 17025599. /wiki/Bibcode_(identifier) ↩

1035. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1036. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1037. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1038. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1039. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1040. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1041. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1042. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1043. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1044. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1045. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

1046. Ballow, A.; P. Linton; D. J. Priour Jr. (2023). "Percolation through voids around toroidal inclusions". Physical Review E. 107 (1): 014902. arXiv:2208.10582. Bibcode:2023PhRvE.107a4902B. doi:10.1103/PhysRevE.107.014902. PMID 36797924. S2CID 251741342. /wiki/ArXiv_(identifier) ↩

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1078. Zakalyukin, R. M.; V. A. Chizhikov (2005). "Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method". Crystallography Reports. 50 (6): 938–948. Bibcode:2005CryRp..50..938Z. doi:10.1134/1.2132400. S2CID 94290876. /wiki/Bibcode_(identifier) ↩

1079. Zakalyukin, R. M.; V. A. Chizhikov (2005). "Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method". Crystallography Reports. 50 (6): 938–948. Bibcode:2005CryRp..50..938Z. doi:10.1134/1.2132400. S2CID 94290876. /wiki/Bibcode_(identifier) ↩

1080. Zakalyukin, R. M.; V. A. Chizhikov (2005). "Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method". Crystallography Reports. 50 (6): 938–948. Bibcode:2005CryRp..50..938Z. doi:10.1134/1.2132400. S2CID 94290876. /wiki/Bibcode_(identifier) ↩

1081. Zakalyukin, R. M.; V. A. Chizhikov (2005). "Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method". Crystallography Reports. 50 (6): 938–948. Bibcode:2005CryRp..50..938Z. doi:10.1134/1.2132400. S2CID 94290876. /wiki/Bibcode_(identifier) ↩

1082. Zakalyukin, R. M.; V. A. Chizhikov (2005). "Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method". Crystallography Reports. 50 (6): 938–948. Bibcode:2005CryRp..50..938Z. doi:10.1134/1.2132400. S2CID 94290876. /wiki/Bibcode_(identifier) ↩

1083. Zakalyukin, R. M.; V. A. Chizhikov (2005). "Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method". Crystallography Reports. 50 (6): 938–948. Bibcode:2005CryRp..50..938Z. doi:10.1134/1.2132400. S2CID 94290876. /wiki/Bibcode_(identifier) ↩

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1088. Grassberger, P. (2017). "Some remarks on drilling percolation". Physical Review E. 95 (1): 010103. arXiv:1611.07939. doi:10.1103/PhysRevE.95.010103. PMID 28208497. S2CID 12476714. http://juser.fz-juelich.de/search?p=id:%22FZJ-2017-07804%22 ↩

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1090. Szczygieł, Bartłomiej; Kamil Kwiatkowski; Maciej Lewenstein; Gerald John Lapeyre, Jr.; Jan Wehr (2016). "Percolation thresholds for discrete-continuous models with nonuniform probabilities of bond formation". Physical Review E. 93 (2): 022127. arXiv:1509.07401. Bibcode:2016PhRvE..93b2127S. doi:10.1103/PhysRevE.93.022127. PMID 26986308. S2CID 18110437. /wiki/ArXiv_(identifier) ↩

1091. Abete, T.; A. de Candia; D. Lairez; A. Coniglio (2004). "Percolation Model for Enzyme Gel Degradation". Physical Review Letters. 93: 228301. arXiv:cond-mat/0402551. doi:10.1103/PhysRevLett.93.228301. /wiki/ArXiv_(identifier) ↩

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1093. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1094. Dall, Jesper; Michael Christensen (2002). "Random geometric graphs". Physical Review E. 66 (1): 016121. arXiv:cond-mat/0203026. Bibcode:2002PhRvE..66a6121D. doi:10.1103/PhysRevE.66.016121. PMID 12241440. S2CID 15193516. /wiki/ArXiv_(identifier) ↩

1095. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1096. Brzeski, Piotr; Kondrat, Grzegorz (2022). "Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous". Journal of Statistical Mechanics: Theory and Experiment. 2022 (5): 053202. doi:10.1088/1742-5468/ac6519. ISSN 1742-5468. https://iopscience.iop.org/article/10.1088/1742-5468/ac6519 ↩

1097. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1098. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056. /wiki/ArXiv_(identifier) ↩

1099. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1100. Jin, Yuliang; Patrick Charbonneau (2014). "Mapping the arrest of the random Lorentz gas onto the dynamical transition of a simple glass former". Physical Review E. 91 (4): 042313. arXiv:1409.0688. Bibcode:2015PhRvE..91d2313J. doi:10.1103/PhysRevE.91.042313. PMID 25974497. S2CID 16117644. /wiki/ArXiv_(identifier) ↩

1101. Jin, Yuliang; Patrick Charbonneau (2014). "Mapping the arrest of the random Lorentz gas onto the dynamical transition of a simple glass former". Physical Review E. 91 (4): 042313. arXiv:1409.0688. Bibcode:2015PhRvE..91d2313J. doi:10.1103/PhysRevE.91.042313. PMID 25974497. S2CID 16117644. /wiki/ArXiv_(identifier) ↩

1102. Charbonneau, Benoit; Patrick Charbonneau; Yi Hu; Zhen Yang (2021). "High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas". Physical Review E. 104 (2): 024137. arXiv:2105.04711. Bibcode:2021PhRvE.104b4137C. doi:10.1103/PhysRevE.104.024137. PMID 34525662. S2CID 234357912. /wiki/ArXiv_(identifier) ↩

1103. Dall, Jesper; Michael Christensen (2002). "Random geometric graphs". Physical Review E. 66 (1): 016121. arXiv:cond-mat/0203026. Bibcode:2002PhRvE..66a6121D. doi:10.1103/PhysRevE.66.016121. PMID 12241440. S2CID 15193516. /wiki/ArXiv_(identifier) ↩

1104. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1105. Brzeski, Piotr; Kondrat, Grzegorz (2022). "Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous". Journal of Statistical Mechanics: Theory and Experiment. 2022 (5): 053202. doi:10.1088/1742-5468/ac6519. ISSN 1742-5468. https://iopscience.iop.org/article/10.1088/1742-5468/ac6519 ↩

1106. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1107. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056. /wiki/ArXiv_(identifier) ↩

1108. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1109. Jin, Yuliang; Patrick Charbonneau (2014). "Mapping the arrest of the random Lorentz gas onto the dynamical transition of a simple glass former". Physical Review E. 91 (4): 042313. arXiv:1409.0688. Bibcode:2015PhRvE..91d2313J. doi:10.1103/PhysRevE.91.042313. PMID 25974497. S2CID 16117644. /wiki/ArXiv_(identifier) ↩

1110. Jin, Yuliang; Patrick Charbonneau (2014). "Mapping the arrest of the random Lorentz gas onto the dynamical transition of a simple glass former". Physical Review E. 91 (4): 042313. arXiv:1409.0688. Bibcode:2015PhRvE..91d2313J. doi:10.1103/PhysRevE.91.042313. PMID 25974497. S2CID 16117644. /wiki/ArXiv_(identifier) ↩

1111. Charbonneau, Benoit; Patrick Charbonneau; Yi Hu; Zhen Yang (2021). "High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas". Physical Review E. 104 (2): 024137. arXiv:2105.04711. Bibcode:2021PhRvE.104b4137C. doi:10.1103/PhysRevE.104.024137. PMID 34525662. S2CID 234357912. /wiki/ArXiv_(identifier) ↩

1112. Dall, Jesper; Michael Christensen (2002). "Random geometric graphs". Physical Review E. 66 (1): 016121. arXiv:cond-mat/0203026. Bibcode:2002PhRvE..66a6121D. doi:10.1103/PhysRevE.66.016121. PMID 12241440. S2CID 15193516. /wiki/ArXiv_(identifier) ↩

1113. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1114. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1115. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056. /wiki/ArXiv_(identifier) ↩

1116. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1117. Jin, Yuliang; Patrick Charbonneau (2014). "Mapping the arrest of the random Lorentz gas onto the dynamical transition of a simple glass former". Physical Review E. 91 (4): 042313. arXiv:1409.0688. Bibcode:2015PhRvE..91d2313J. doi:10.1103/PhysRevE.91.042313. PMID 25974497. S2CID 16117644. /wiki/ArXiv_(identifier) ↩

1118. Jin, Yuliang; Patrick Charbonneau (2014). "Mapping the arrest of the random Lorentz gas onto the dynamical transition of a simple glass former". Physical Review E. 91 (4): 042313. arXiv:1409.0688. Bibcode:2015PhRvE..91d2313J. doi:10.1103/PhysRevE.91.042313. PMID 25974497. S2CID 16117644. /wiki/ArXiv_(identifier) ↩

1119. Charbonneau, Benoit; Patrick Charbonneau; Yi Hu; Zhen Yang (2021). "High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas". Physical Review E. 104 (2): 024137. arXiv:2105.04711. Bibcode:2021PhRvE.104b4137C. doi:10.1103/PhysRevE.104.024137. PMID 34525662. S2CID 234357912. /wiki/ArXiv_(identifier) ↩

1120. Dall, Jesper; Michael Christensen (2002). "Random geometric graphs". Physical Review E. 66 (1): 016121. arXiv:cond-mat/0203026. Bibcode:2002PhRvE..66a6121D. doi:10.1103/PhysRevE.66.016121. PMID 12241440. S2CID 15193516. /wiki/ArXiv_(identifier) ↩

1121. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1122. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1123. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056. /wiki/ArXiv_(identifier) ↩

1124. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1125. Charbonneau, Benoit; Patrick Charbonneau; Yi Hu; Zhen Yang (2021). "High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas". Physical Review E. 104 (2): 024137. arXiv:2105.04711. Bibcode:2021PhRvE.104b4137C. doi:10.1103/PhysRevE.104.024137. PMID 34525662. S2CID 234357912. /wiki/ArXiv_(identifier) ↩

1126. Dall, Jesper; Michael Christensen (2002). "Random geometric graphs". Physical Review E. 66 (1): 016121. arXiv:cond-mat/0203026. Bibcode:2002PhRvE..66a6121D. doi:10.1103/PhysRevE.66.016121. PMID 12241440. S2CID 15193516. /wiki/ArXiv_(identifier) ↩

1127. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1128. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1129. Charbonneau, Benoit; Patrick Charbonneau; Yi Hu; Zhen Yang (2021). "High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas". Physical Review E. 104 (2): 024137. arXiv:2105.04711. Bibcode:2021PhRvE.104b4137C. doi:10.1103/PhysRevE.104.024137. PMID 34525662. S2CID 234357912. /wiki/ArXiv_(identifier) ↩

1130. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1131. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1132. Charbonneau, Benoit; Patrick Charbonneau; Yi Hu; Zhen Yang (2021). "High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas". Physical Review E. 104 (2): 024137. arXiv:2105.04711. Bibcode:2021PhRvE.104b4137C. doi:10.1103/PhysRevE.104.024137. PMID 34525662. S2CID 234357912. /wiki/ArXiv_(identifier) ↩

1133. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1134. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1135. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1136. Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. PMID 22920102. S2CID 13188197. /wiki/ArXiv_(identifier) ↩

1137. Kirkpatrick, Scott (1976). "Percolation phenomena in higher dimensions: Approach to the mean-field limit". Physical Review Letters. 36 (2): 69–72. Bibcode:1976PhRvL..36...69K. doi:10.1103/PhysRevLett.36.69. /wiki/Bibcode_(identifier) ↩

1138. Gaunt, D. S.; Sykes, M. F.; Ruskin, Heather (1976). "Percolation processes in d-dimensions". J. Phys. A: Math. Gen. 9 (11): 1899–1911. Bibcode:1976JPhA....9.1899G. doi:10.1088/0305-4470/9/11/015. /wiki/Bibcode_(identifier) ↩

1139. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1140. Paul, Gerald; Robert M. Ziff; H. Eugene Stanley (2001). "Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions". Physical Review E. 64 (2): 8. arXiv:cond-mat/0101136. Bibcode:2001PhRvE..64b6115P. doi:10.1103/PhysRevE.64.026115. PMID 11497659. S2CID 18271196. /wiki/ArXiv_(identifier) ↩

1141. Ballesteros, H. G.; L. A. Fernández; V. Martín-Mayor; A. Muñoz Sudupe; G. Parisi; J. J. Ruiz-Lorenzo (1997). "Measures of critical exponents in the four dimensional site percolation". Phys. Lett. B. 400 (3–4): 346–351. arXiv:hep-lat/9612024. Bibcode:1997PhLB..400..346B. doi:10.1016/S0370-2693(97)00337-7. S2CID 10242417. /wiki/ArXiv_(identifier) ↩

1142. Kotwica, M.; P. Gronek; K. Malarz (2019). "Efficient space virtualisation for Hoshen–Kopelman algorithm". International Journal of Modern Physics C. 30 (8): 1950055–1950099. arXiv:1803.09504. Bibcode:2019IJMPC..3050055K. doi:10.1142/S0129183119500554. S2CID 4418563. /wiki/ArXiv_(identifier) ↩

1143. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056. /wiki/ArXiv_(identifier) ↩

1144. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1145. Harris, A. B.; Fisch, R. (1977). "Critical Behavior of Random Resistor Networks". Physical Review Letters. 38 (15): 796–799. Bibcode:1977PhRvL..38..796H. doi:10.1103/PhysRevLett.38.796. https://link.aps.org/doi/10.1103/PhysRevLett.38.796 ↩

1146. Adler, Joan; Yigal Meir; Amnon Aharony; A. B. Harris; Lior Klein (1990). "Low-Concentration Series in General Dimension". Journal of Statistical Physics. 58 (3/4): 511–538. Bibcode:1990JSP....58..511A. doi:10.1007/BF01112760. S2CID 122109020. /wiki/Bibcode_(identifier) ↩

1147. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1148. Paul, Gerald; Robert M. Ziff; H. Eugene Stanley (2001). "Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions". Physical Review E. 64 (2): 8. arXiv:cond-mat/0101136. Bibcode:2001PhRvE..64b6115P. doi:10.1103/PhysRevE.64.026115. PMID 11497659. S2CID 18271196. /wiki/ArXiv_(identifier) ↩

1149. Dammer, Stephan M; Haye Hinrichsen (2004). "Spreading with immunization in high dimensions". Journal of Statistical Mechanics: Theory and Experiment. 2004 (7): P07011. arXiv:cond-mat/0405577. Bibcode:2004JSMTE..07..011D. doi:10.1088/1742-5468/2004/07/P07011. S2CID 118981083. /wiki/ArXiv_(identifier) ↩

1150. Xun, Zhipeng (2020). "Precise bond percolation thresholds on several four-dimensional lattices". Physical Review Research. 2 (1): 013067. arXiv:1910.11408. Bibcode:2020PhRvR...2a3067X. doi:10.1103/PhysRevResearch.2.013067. S2CID 204915841. /wiki/ArXiv_(identifier) ↩

1151. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1152. Kirkpatrick, Scott (1976). "Percolation phenomena in higher dimensions: Approach to the mean-field limit". Physical Review Letters. 36 (2): 69–72. Bibcode:1976PhRvL..36...69K. doi:10.1103/PhysRevLett.36.69. /wiki/Bibcode_(identifier) ↩

1153. Gaunt, D. S.; Sykes, M. F.; Ruskin, Heather (1976). "Percolation processes in d-dimensions". J. Phys. A: Math. Gen. 9 (11): 1899–1911. Bibcode:1976JPhA....9.1899G. doi:10.1088/0305-4470/9/11/015. /wiki/Bibcode_(identifier) ↩

1154. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1155. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056. /wiki/ArXiv_(identifier) ↩

1156. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1157. Harris, A. B.; Fisch, R. (1977). "Critical Behavior of Random Resistor Networks". Physical Review Letters. 38 (15): 796–799. Bibcode:1977PhRvL..38..796H. doi:10.1103/PhysRevLett.38.796. https://link.aps.org/doi/10.1103/PhysRevLett.38.796 ↩

1158. Gaunt, D. S.; Ruskin, Heather (1978). "Bond percolation processes in d-dimensions". J. Phys. A: Math. Gen. 11 (7): 1369. Bibcode:1978JPhA...11.1369G. doi:10.1088/0305-4470/11/7/025. /wiki/Bibcode_(identifier) ↩

1159. Adler, Joan; Yigal Meir; Amnon Aharony; A. B. Harris; Lior Klein (1990). "Low-Concentration Series in General Dimension". Journal of Statistical Physics. 58 (3/4): 511–538. Bibcode:1990JSP....58..511A. doi:10.1007/BF01112760. S2CID 122109020. /wiki/Bibcode_(identifier) ↩

1160. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1161. Dammer, Stephan M; Haye Hinrichsen (2004). "Spreading with immunization in high dimensions". Journal of Statistical Mechanics: Theory and Experiment. 2004 (7): P07011. arXiv:cond-mat/0405577. Bibcode:2004JSMTE..07..011D. doi:10.1088/1742-5468/2004/07/P07011. S2CID 118981083. /wiki/ArXiv_(identifier) ↩

1162. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1163. Kirkpatrick, Scott (1976). "Percolation phenomena in higher dimensions: Approach to the mean-field limit". Physical Review Letters. 36 (2): 69–72. Bibcode:1976PhRvL..36...69K. doi:10.1103/PhysRevLett.36.69. /wiki/Bibcode_(identifier) ↩

1164. Gaunt, D. S.; Sykes, M. F.; Ruskin, Heather (1976). "Percolation processes in d-dimensions". J. Phys. A: Math. Gen. 9 (11): 1899–1911. Bibcode:1976JPhA....9.1899G. doi:10.1088/0305-4470/9/11/015. /wiki/Bibcode_(identifier) ↩

1165. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1166. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056. /wiki/ArXiv_(identifier) ↩

1167. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1168. Harris, A. B.; Fisch, R. (1977). "Critical Behavior of Random Resistor Networks". Physical Review Letters. 38 (15): 796–799. Bibcode:1977PhRvL..38..796H. doi:10.1103/PhysRevLett.38.796. https://link.aps.org/doi/10.1103/PhysRevLett.38.796 ↩

1169. Adler, Joan; Yigal Meir; Amnon Aharony; A. B. Harris (1990). "Series Study of Percolation Moments in General Dimension". Physical Review B. 41 (13): 9183–9206. Bibcode:1990PhRvB..41.9183A. doi:10.1103/PhysRevB.41.9183. PMID 9993262. https://repository.upenn.edu/physics_papers/368 ↩

1170. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1171. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1172. Adler, Joan; Yigal Meir; Amnon Aharony; A. B. Harris (1990). "Series Study of Percolation Moments in General Dimension". Physical Review B. 41 (13): 9183–9206. Bibcode:1990PhRvB..41.9183A. doi:10.1103/PhysRevB.41.9183. PMID 9993262. https://repository.upenn.edu/physics_papers/368 ↩

1173. Stauffer, Dietrich; Robert M. Ziff (1999). "Reexamination of Seven-Dimensional Site Percolation Thresholds". International Journal of Modern Physics C. 11 (1): 205–209. arXiv:cond-mat/9911090. Bibcode:2000IJMPC..11..205S. doi:10.1142/S0129183100000183. S2CID 119362011. /wiki/ArXiv_(identifier) ↩

1174. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1175. Koza, Zbigniew; Jakub Poła (2016). "From discrete to continuous percolation in dimensions 3 to 7". Journal of Statistical Mechanics: Theory and Experiment. 2016 (10): 103206. arXiv:1606.08050. Bibcode:2016JSMTE..10.3206K. doi:10.1088/1742-5468/2016/10/103206. S2CID 118580056. /wiki/ArXiv_(identifier) ↩

1176. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1177. Harris, A. B.; Fisch, R. (1977). "Critical Behavior of Random Resistor Networks". Physical Review Letters. 38 (15): 796–799. Bibcode:1977PhRvL..38..796H. doi:10.1103/PhysRevLett.38.796. https://link.aps.org/doi/10.1103/PhysRevLett.38.796 ↩

1178. Adler, Joan; Yigal Meir; Amnon Aharony; A. B. Harris (1990). "Series Study of Percolation Moments in General Dimension". Physical Review B. 41 (13): 9183–9206. Bibcode:1990PhRvB..41.9183A. doi:10.1103/PhysRevB.41.9183. PMID 9993262. https://repository.upenn.edu/physics_papers/368 ↩

1179. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1180. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1181. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1182. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1183. Adler, Joan; Yigal Meir; Amnon Aharony; A. B. Harris (1990). "Series Study of Percolation Moments in General Dimension". Physical Review B. 41 (13): 9183–9206. Bibcode:1990PhRvB..41.9183A. doi:10.1103/PhysRevB.41.9183. PMID 9993262. https://repository.upenn.edu/physics_papers/368 ↩

1184. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1185. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1186. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1187. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1188. Adler, Joan; Yigal Meir; Amnon Aharony; A. B. Harris (1990). "Series Study of Percolation Moments in General Dimension". Physical Review B. 41 (13): 9183–9206. Bibcode:1990PhRvB..41.9183A. doi:10.1103/PhysRevB.41.9183. PMID 9993262. https://repository.upenn.edu/physics_papers/368 ↩

1189. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1190. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1191. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1192. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1193. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1194. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1195. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1196. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1197. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1198. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1199. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1200. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1201. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1202. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1203. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1204. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1205. Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E. 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. PMID 12689126. S2CID 43707822. /wiki/ArXiv_(identifier) ↩

1206. Mertens, Stephan; Christopher Moore (2018). "Percolation Thresholds and Fisher Exponents in Hypercubic Lattices". Physical Review E. 98 (2): 022120. arXiv:1806.08067. Bibcode:2018PhRvE..98b2120M. doi:10.1103/PhysRevE.98.022120. PMID 30253462. S2CID 52821851. /wiki/ArXiv_(identifier) ↩

1207. Gaunt, D. S.; Sykes, M. F.; Ruskin, Heather (1976). "Percolation processes in d-dimensions". J. Phys. A: Math. Gen. 9 (11): 1899–1911. Bibcode:1976JPhA....9.1899G. doi:10.1088/0305-4470/9/11/015. /wiki/Bibcode_(identifier) ↩

1208. Gaunt, D. S.; Ruskin, Heather (1978). "Bond percolation processes in d-dimensions". J. Phys. A: Math. Gen. 11 (7): 1369. Bibcode:1978JPhA...11.1369G. doi:10.1088/0305-4470/11/7/025. /wiki/Bibcode_(identifier) ↩

1209. Mertens, Stephan; Moore, Christopher (2018). "Series Expansion of Critical Densities for Percolation on ℤd". J. Phys. A: Math. Theor. 51 (47): 475001. arXiv:1805.02701. doi:10.1088/1751-8121/aae65c. S2CID 119399128. /wiki/ArXiv_(identifier) ↩

1210. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1211. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1212. van der Marck, Steven C. (1998). "Site percolation and random walks on d-dimensional Kagome lattices". Journal of Physics A. 31 (15): 3449–3460. arXiv:cond-mat/9801112. Bibcode:1998JPhA...31.3449V. doi:10.1088/0305-4470/31/15/010. S2CID 18989583. /wiki/ArXiv_(identifier) ↩

1213. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1214. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1215. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1216. Xun, Zhipeng (2020). "Precise bond percolation thresholds on several four-dimensional lattices". Physical Review Research. 2 (1): 013067. arXiv:1910.11408. Bibcode:2020PhRvR...2a3067X. doi:10.1103/PhysRevResearch.2.013067. S2CID 204915841. /wiki/ArXiv_(identifier) ↩

1217. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1218. Kotwica, M.; P. Gronek; K. Malarz (2019). "Efficient space virtualisation for Hoshen–Kopelman algorithm". International Journal of Modern Physics C. 30 (8): 1950055–1950099. arXiv:1803.09504. Bibcode:2019IJMPC..3050055K. doi:10.1142/S0129183119500554. S2CID 4418563. /wiki/ArXiv_(identifier) ↩

1219. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1220. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1221. Xun, Zhipeng (2020). "Precise bond percolation thresholds on several four-dimensional lattices". Physical Review Research. 2 (1): 013067. arXiv:1910.11408. Bibcode:2020PhRvR...2a3067X. doi:10.1103/PhysRevResearch.2.013067. S2CID 204915841. /wiki/ArXiv_(identifier) ↩

1222. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1223. Kotwica, M.; P. Gronek; K. Malarz (2019). "Efficient space virtualisation for Hoshen–Kopelman algorithm". International Journal of Modern Physics C. 30 (8): 1950055–1950099. arXiv:1803.09504. Bibcode:2019IJMPC..3050055K. doi:10.1142/S0129183119500554. S2CID 4418563. /wiki/ArXiv_(identifier) ↩

1224. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1225. Xun, Zhipeng (2020). "Precise bond percolation thresholds on several four-dimensional lattices". Physical Review Research. 2 (1): 013067. arXiv:1910.11408. Bibcode:2020PhRvR...2a3067X. doi:10.1103/PhysRevResearch.2.013067. S2CID 204915841. /wiki/ArXiv_(identifier) ↩

1226. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1227. Kotwica, M.; P. Gronek; K. Malarz (2019). "Efficient space virtualisation for Hoshen–Kopelman algorithm". International Journal of Modern Physics C. 30 (8): 1950055–1950099. arXiv:1803.09504. Bibcode:2019IJMPC..3050055K. doi:10.1142/S0129183119500554. S2CID 4418563. /wiki/ArXiv_(identifier) ↩

1228. Kotwica, M.; P. Gronek; K. Malarz (2019). "Efficient space virtualisation for Hoshen–Kopelman algorithm". International Journal of Modern Physics C. 30 (8): 1950055–1950099. arXiv:1803.09504. Bibcode:2019IJMPC..3050055K. doi:10.1142/S0129183119500554. S2CID 4418563. /wiki/ArXiv_(identifier) ↩

1229. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1230. Kotwica, M.; P. Gronek; K. Malarz (2019). "Efficient space virtualisation for Hoshen–Kopelman algorithm". International Journal of Modern Physics C. 30 (8): 1950055–1950099. arXiv:1803.09504. Bibcode:2019IJMPC..3050055K. doi:10.1142/S0129183119500554. S2CID 4418563. /wiki/ArXiv_(identifier) ↩

1231. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1232. Kotwica, M.; P. Gronek; K. Malarz (2019). "Efficient space virtualisation for Hoshen–Kopelman algorithm". International Journal of Modern Physics C. 30 (8): 1950055–1950099. arXiv:1803.09504. Bibcode:2019IJMPC..3050055K. doi:10.1142/S0129183119500554. S2CID 4418563. /wiki/ArXiv_(identifier) ↩

1233. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1234. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1235. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1236. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1237. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1238. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1239. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1240. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1241. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1242. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1243. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1244. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1245. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1246. Zhao, Pengyu; Jinhong Yan; Zhipeng Xun; Dapeng Hao; Robert M. Ziff (2022). "Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods". Journal of Statistical Mechanics: Theory and Experiment. 2022 (3): 033202. arXiv:2109.11195. Bibcode:2022JSMTE2022c3202Z. doi:10.1088/1742-5468/ac52a8. S2CID 237605083. /wiki/ArXiv_(identifier) ↩

1247. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1248. van der Marck, Steven C. (1998). "Site percolation and random walks on d-dimensional Kagome lattices". Journal of Physics A. 31 (15): 3449–3460. arXiv:cond-mat/9801112. Bibcode:1998JPhA...31.3449V. doi:10.1088/0305-4470/31/15/010. S2CID 18989583. /wiki/ArXiv_(identifier) ↩

1249. van der Marck, Steven C. (1998). "Site percolation and random walks on d-dimensional Kagome lattices". Journal of Physics A. 31 (15): 3449–3460. arXiv:cond-mat/9801112. Bibcode:1998JPhA...31.3449V. doi:10.1088/0305-4470/31/15/010. S2CID 18989583. /wiki/ArXiv_(identifier) ↩

1250. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1251. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1252. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1253. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1254. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1255. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1256. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1257. Löbl, Matthias C. (2024). "Loss-tolerant architecture for quantum computing with quantum emitters". Quantum. 8: 1302. arXiv:2304.03796. doi:10.22331/q-2024-03-28-1302. /wiki/ArXiv_(identifier) ↩

1258. Löbl, Matthias C. (2024). "Loss-tolerant architecture for quantum computing with quantum emitters". Quantum. 8: 1302. arXiv:2304.03796. doi:10.22331/q-2024-03-28-1302. /wiki/ArXiv_(identifier) ↩

1259. van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C. 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. S2CID 119097158. /wiki/ArXiv_(identifier) ↩

1260. van der Marck, Steven C. (1998). "Site percolation and random walks on d-dimensional Kagome lattices". Journal of Physics A. 31 (15): 3449–3460. arXiv:cond-mat/9801112. Bibcode:1998JPhA...31.3449V. doi:10.1088/0305-4470/31/15/010. S2CID 18989583. /wiki/ArXiv_(identifier) ↩

1261. van der Marck, Steven C. (1998). "Site percolation and random walks on d-dimensional Kagome lattices". Journal of Physics A. 31 (15): 3449–3460. arXiv:cond-mat/9801112. Bibcode:1998JPhA...31.3449V. doi:10.1088/0305-4470/31/15/010. S2CID 18989583. /wiki/ArXiv_(identifier) ↩

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1264. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

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1267. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

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1269. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

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1271. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1272. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1273. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1274. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1275. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1276. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1277. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1278. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1279. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1280. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1281. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1282. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1283. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1284. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1285. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1286. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1287. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

1288. Hu, Yi; Patrick Charbonneau (2021). "Percolation thresholds on high-dimensional Dn and E8-related lattices". Physical Review E. 103 (6): 062115. arXiv:2102.09682. Bibcode:2021PhRvE.103f2115H. doi:10.1103/PhysRevE.103.062115. PMID 34271715. S2CID 231979212. /wiki/ArXiv_(identifier) ↩

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1297. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1298. Lopez, Jorge H.; J. M. Schwarz (2017). "Constraint percolation on hyperbolic lattices". Physical Review E. 96 (5): 052108. arXiv:1512.05404. Bibcode:2017PhRvE..96e2108L. doi:10.1103/PhysRevE.96.052108. PMID 29347694. S2CID 44770310. /wiki/Jennifer_Schwarz ↩

1299. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1300. Lopez, Jorge H.; J. M. Schwarz (2017). "Constraint percolation on hyperbolic lattices". Physical Review E. 96 (5): 052108. arXiv:1512.05404. Bibcode:2017PhRvE..96e2108L. doi:10.1103/PhysRevE.96.052108. PMID 29347694. S2CID 44770310. /wiki/Jennifer_Schwarz ↩

1301. Baek, S.K.; Petter Minnhagen; Beom Jun Kim (2009). "Percolation on hyperbolic lattices". Physical Review E. 79 (1): 011124. arXiv:0901.0483. Bibcode:2009PhRvE..79a1124B. doi:10.1103/PhysRevE.79.011124. PMID 19257018. S2CID 29468086. /wiki/ArXiv_(identifier) ↩

1302. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1303. Baek, S.K.; Petter Minnhagen; Beom Jun Kim (2009). "Percolation on hyperbolic lattices". Physical Review E. 79 (1): 011124. arXiv:0901.0483. Bibcode:2009PhRvE..79a1124B. doi:10.1103/PhysRevE.79.011124. PMID 19257018. S2CID 29468086. /wiki/ArXiv_(identifier) ↩

1304. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1305. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1306. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1307. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1308. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1309. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1310. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1311. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1312. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1313. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1314. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1315. Baek, S.K.; Petter Minnhagen; Beom Jun Kim (2009). "Percolation on hyperbolic lattices". Physical Review E. 79 (1): 011124. arXiv:0901.0483. Bibcode:2009PhRvE..79a1124B. doi:10.1103/PhysRevE.79.011124. PMID 19257018. S2CID 29468086. /wiki/ArXiv_(identifier) ↩

1316. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1317. Baek, S.K.; Petter Minnhagen; Beom Jun Kim (2009). "Percolation on hyperbolic lattices". Physical Review E. 79 (1): 011124. arXiv:0901.0483. Bibcode:2009PhRvE..79a1124B. doi:10.1103/PhysRevE.79.011124. PMID 19257018. S2CID 29468086. /wiki/ArXiv_(identifier) ↩

1318. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1319. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1320. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1321. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1322. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1323. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1324. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1325. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1326. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1327. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1328. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1329. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1330. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1331. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1332. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1333. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1334. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1335. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1336. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

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1338. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1339. Gu, Hang; Robert M. Ziff (2012). "Crossing on hyperbolic lattices". Physical Review E. 85 (5): 051141. arXiv:1111.5626. Bibcode:2012PhRvE..85e1141G. doi:10.1103/PhysRevE.85.051141. PMID 23004737. S2CID 7141649. /wiki/ArXiv_(identifier) ↩

1340. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

1341. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

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1345. Gu, Hang; Robert M. Ziff (2012). "Crossing on hyperbolic lattices". Physical Review E. 85 (5): 051141. arXiv:1111.5626. Bibcode:2012PhRvE..85e1141G. doi:10.1103/PhysRevE.85.051141. PMID 23004737. S2CID 7141649. /wiki/ArXiv_(identifier) ↩

1346. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

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1348. Gu, Hang; Robert M. Ziff (2012). "Crossing on hyperbolic lattices". Physical Review E. 85 (5): 051141. arXiv:1111.5626. Bibcode:2012PhRvE..85e1141G. doi:10.1103/PhysRevE.85.051141. PMID 23004737. S2CID 7141649. /wiki/ArXiv_(identifier) ↩

1349. Mertens, Stephan; Cristopher Moore (2017). "Percolation thresholds in hyperbolic lattices". Physical Review E. 96 (4): 042116. arXiv:1708.05876. Bibcode:2017PhRvE..96d2116M. doi:10.1103/PhysRevE.96.042116. PMID 29347529. S2CID 39025690. /wiki/ArXiv_(identifier) ↩

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1351. Baek, S.K.; Petter Minnhagen; Beom Jun Kim (2009). "Percolation on hyperbolic lattices". Physical Review E. 79 (1): 011124. arXiv:0901.0483. Bibcode:2009PhRvE..79a1124B. doi:10.1103/PhysRevE.79.011124. PMID 19257018. S2CID 29468086. /wiki/ArXiv_(identifier) ↩

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1408. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1409. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1410. Blease, J. (1977). "Series expansions for the directed-bond percolation problem". J. Phys. C: Solid State Phys. 10 (7): 917–924. Bibcode:1977JPhC...10..917B. doi:10.1088/0022-3719/10/7/003. /wiki/Bibcode_(identifier) ↩

1411. Grassberger, P. (2009). "Local persistence in directed percolation". Journal of Statistical Mechanics: Theory and Experiment. 2009 (8): P08021. arXiv:0907.4021. Bibcode:2009JSMTE..08..021G. doi:10.1088/1742-5468/2009/08/P08021. S2CID 119236556. /wiki/ArXiv_(identifier) ↩

1412. Dammer, Stephan M; Haye Hinrichsen (2004). "Spreading with immunization in high dimensions". Journal of Statistical Mechanics: Theory and Experiment. 2004 (7): P07011. arXiv:cond-mat/0405577. Bibcode:2004JSMTE..07..011D. doi:10.1088/1742-5468/2004/07/P07011. S2CID 118981083. /wiki/ArXiv_(identifier) ↩

1413. Lübeck, S.; R. D. Willmann (2004). "Universal scaling behavior of directed percolation around the upper critical dimension". J. Stat. Phys. 115 (5–6): 1231–1250. arXiv:cond-mat/0401395. Bibcode:2004JSP...115.1231L. CiteSeerX 10.1.1.310.8700. doi:10.1023/B:JOSS.0000028059.24904.3b. S2CID 16267627. /wiki/ArXiv_(identifier) ↩

1414. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1415. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1416. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1417. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1418. Blease, J. (1977). "Series expansions for the directed-bond percolation problem". J. Phys. C: Solid State Phys. 10 (7): 917–924. Bibcode:1977JPhC...10..917B. doi:10.1088/0022-3719/10/7/003. /wiki/Bibcode_(identifier) ↩

1419. Grassberger, P. (2009). "Local persistence in directed percolation". Journal of Statistical Mechanics: Theory and Experiment. 2009 (8): P08021. arXiv:0907.4021. Bibcode:2009JSMTE..08..021G. doi:10.1088/1742-5468/2009/08/P08021. S2CID 119236556. /wiki/ArXiv_(identifier) ↩

1420. Grassberger, Peter (2009). "Logarithmic corrections in (4 + 1)-dimensional directed percolation". Physical Review E. 79 (5): 052104. arXiv:0904.0804. Bibcode:2009PhRvE..79e2104G. doi:10.1103/PhysRevE.79.052104. PMID 19518501. S2CID 23876626. /wiki/ArXiv_(identifier) ↩

1421. Dammer, Stephan M; Haye Hinrichsen (2004). "Spreading with immunization in high dimensions". Journal of Statistical Mechanics: Theory and Experiment. 2004 (7): P07011. arXiv:cond-mat/0405577. Bibcode:2004JSMTE..07..011D. doi:10.1088/1742-5468/2004/07/P07011. S2CID 118981083. /wiki/ArXiv_(identifier) ↩

1422. Lübeck, S.; R. D. Willmann (2004). "Universal scaling behavior of directed percolation around the upper critical dimension". J. Stat. Phys. 115 (5–6): 1231–1250. arXiv:cond-mat/0401395. Bibcode:2004JSP...115.1231L. CiteSeerX 10.1.1.310.8700. doi:10.1023/B:JOSS.0000028059.24904.3b. S2CID 16267627. /wiki/ArXiv_(identifier) ↩

1423. Grassberger, Peter (2009). "Logarithmic corrections in (4 + 1)-dimensional directed percolation". Physical Review E. 79 (5): 052104. arXiv:0904.0804. Bibcode:2009PhRvE..79e2104G. doi:10.1103/PhysRevE.79.052104. PMID 19518501. S2CID 23876626. /wiki/ArXiv_(identifier) ↩

1424. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1425. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1426. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1427. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1428. Blease, J. (1977). "Series expansions for the directed-bond percolation problem". J. Phys. C: Solid State Phys. 10 (7): 917–924. Bibcode:1977JPhC...10..917B. doi:10.1088/0022-3719/10/7/003. /wiki/Bibcode_(identifier) ↩

1429. Grassberger, P. (2009). "Local persistence in directed percolation". Journal of Statistical Mechanics: Theory and Experiment. 2009 (8): P08021. arXiv:0907.4021. Bibcode:2009JSMTE..08..021G. doi:10.1088/1742-5468/2009/08/P08021. S2CID 119236556. /wiki/ArXiv_(identifier) ↩

1430. Dammer, Stephan M; Haye Hinrichsen (2004). "Spreading with immunization in high dimensions". Journal of Statistical Mechanics: Theory and Experiment. 2004 (7): P07011. arXiv:cond-mat/0405577. Bibcode:2004JSMTE..07..011D. doi:10.1088/1742-5468/2004/07/P07011. S2CID 118981083. /wiki/ArXiv_(identifier) ↩

1431. Lübeck, S.; R. D. Willmann (2004). "Universal scaling behavior of directed percolation around the upper critical dimension". J. Stat. Phys. 115 (5–6): 1231–1250. arXiv:cond-mat/0401395. Bibcode:2004JSP...115.1231L. CiteSeerX 10.1.1.310.8700. doi:10.1023/B:JOSS.0000028059.24904.3b. S2CID 16267627. /wiki/ArXiv_(identifier) ↩

1432. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1433. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1434. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1435. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1436. Blease, J. (1977). "Series expansions for the directed-bond percolation problem". J. Phys. C: Solid State Phys. 10 (7): 917–924. Bibcode:1977JPhC...10..917B. doi:10.1088/0022-3719/10/7/003. /wiki/Bibcode_(identifier) ↩

1437. Grassberger, P. (2009). "Local persistence in directed percolation". Journal of Statistical Mechanics: Theory and Experiment. 2009 (8): P08021. arXiv:0907.4021. Bibcode:2009JSMTE..08..021G. doi:10.1088/1742-5468/2009/08/P08021. S2CID 119236556. /wiki/ArXiv_(identifier) ↩

1438. Dammer, Stephan M; Haye Hinrichsen (2004). "Spreading with immunization in high dimensions". Journal of Statistical Mechanics: Theory and Experiment. 2004 (7): P07011. arXiv:cond-mat/0405577. Bibcode:2004JSMTE..07..011D. doi:10.1088/1742-5468/2004/07/P07011. S2CID 118981083. /wiki/ArXiv_(identifier) ↩

1439. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1440. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1441. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1442. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1443. Blease, J. (1977). "Series expansions for the directed-bond percolation problem". J. Phys. C: Solid State Phys. 10 (7): 917–924. Bibcode:1977JPhC...10..917B. doi:10.1088/0022-3719/10/7/003. /wiki/Bibcode_(identifier) ↩

1444. Grassberger, P. (2009). "Local persistence in directed percolation". Journal of Statistical Mechanics: Theory and Experiment. 2009 (8): P08021. arXiv:0907.4021. Bibcode:2009JSMTE..08..021G. doi:10.1088/1742-5468/2009/08/P08021. S2CID 119236556. /wiki/ArXiv_(identifier) ↩

1445. Dammer, Stephan M; Haye Hinrichsen (2004). "Spreading with immunization in high dimensions". Journal of Statistical Mechanics: Theory and Experiment. 2004 (7): P07011. arXiv:cond-mat/0405577. Bibcode:2004JSMTE..07..011D. doi:10.1088/1742-5468/2004/07/P07011. S2CID 118981083. /wiki/ArXiv_(identifier) ↩

1446. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1447. Wang, Junfeng; Zongzheng Zhou; Qingquan Liu; Timothy M. Garoni; Youjin Deng (2013). "A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions". Physical Review E. 88 (4): 042102. arXiv:1201.3006. Bibcode:2013PhRvE..88d2102W. doi:10.1103/PhysRevE.88.042102. PMID 24229111. S2CID 43011467. /wiki/ArXiv_(identifier) ↩

1448. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1449. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1450. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1451. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1452. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1453. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1454. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1455. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1456. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1457. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1458. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

1459. Soares, Danyel J. B.; José S Andrade Jr; Hans J. Herrmann (2006). "Precise calculation of the threshold of various directed percolation models on a square lattice". J. Phys. A: Math. Gen. 38 (21): L413 – L415. arXiv:cond-mat/0503408. doi:10.1088/0305-4470/38/21/L06. /wiki/ArXiv_(identifier) ↩

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