Percolation threshold

Percolation threshold is a mathematical concept related to percolation theory, which is the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability p, or more generally a critical surface for a group of parameters p1, p2, ..., such that infinite connectivity (percolation) first occurs.

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 appears, 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).

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.[1] 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.

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.

The notation such as (4,82) comes from Grünbaum and Shephard,[2] 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.

Thresholds on Archimedean lattices

This is a picture of the 11 Archimedean Lattices or uniform tilings, in which all polygons are regular and each vertex is surrounded by the same sequence of polygons. The notation (34, 6) for example means that every vertex is surrounded by four triangles and one hexagon. Drawings from .[3] See also Uniform Tilings.

Lattice z Site Percolation Threshold Bond Percolation Threshold
3-12 or (3, 122 ) 3 3 0.807900764... = (1 - 2 sin (π/18))1/2[4] 0.7404207988509(8),[5][6] 0.740420800(2),[7] 0.74042195(80),[8] 0.74042077(2)[9]
cross (4, 6, 12) 3 3 0.7478008(2),[5] 0.747806(4)[4] 0.6937314(1),[5] 0.69373383(72)[8]
square octagon, bathroom tile, 4-8, truncated square

(4, 82)

 3 3 0.7297232(5),[5] 0.729724(3)[4] 0.6768031269(6),[5] 0.67680232(63),[8] 0.6768 [10]
honeycomb (63) 3 3 0.6962(6),[11] 0.697040230(5),[5] 0.6970402(1),[12] 0.6970413(10),[13] 0.697043(3),[4] 0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=0[14]
kagome (3, 6, 3, 6) 4 4 0.652703645... = 1 - 2 sin(π/18)[14] 0.524404978(5),[9] 0.52440499(2),[12] 0.52440572...,[15] 0.52440500(1),[7] 0.52440516(10),[13] 0.5244053(3),[16]

0.524404999173(3),[5][6] 0.524404999167439(4)[17]

ruby,[18] rhombitrihexagonal (3, 4, 6, 4) 4 4 0.62181207(7),[5] 0.621819(3)[4] 0.5248311(1),[5] 0.52483258(53)[8]
square (44) 4 4 0.59274605079210(2),[17] 0.59274601(2),[5] 0.59274605095(15),[19] 0.59274621(13),[20] 0.59274621(33),[21] 0.59274598(4),[22][23] 0.59274605(3),[12] 0.593(1),[24]

0.591(1),[25] 0.569(13)[26]

 1/2
snub hexagonal, maple leaf [27] (34,6 ) 5 5 0.579498(3)[4] 0.43432764(3),[5] 0.43430621(50)[8]
snub square, puzzle (32, 4, 3, 4 ) 5 5 0.550806(3)[4] 0.4141378476 (7),[5] 0.41413743(46)[8]
frieze, (33, 42) 5 5 0.550213(3),[4] 0.5502(8)[28] 0.41964044(1),[5] 0.41964191(43),[8] 0.4196(6) [28]
triangular (36) 6 6 1/2 0.347296355... = 2 sin (π/18), 1+ p3-3p=0[14]

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

Square lattice with complex neighborhoods

Lattice z Site Percolation Threshold Bond Percolation Threshold
square: 3N, 4N, 6N 4 0.592...[29][30]
square: 3N+2N, 4N+3N, 6N+4N 8 0.407...[29][30][31]
square: 4N+2N 8 0.337...[29][30]
square: 6N+3N 8 0.337...[30]
square: 5N 8 0.270...[30]
square: 6N+2N 8 0.277...[30]
square: 4N+3N+2N 12 0.288...[29][30]
square: 6N+4N+3N 12 0.288...[30]
square: 5N+2N 12 0.236...[30]
square: 5N+3N 12 0.225...[30]
square: 5N+4N 12 0.221...[30]
square: 6N+3N+2N 12 0.240...[30]
square: 6N+4N+2N 12 0.233...[30]
square: 6N+5N 12 0.199...[30]
square: 5N+3N+2N 16 0.219...[30]
square: 5N+4N+2N 16 0.208...[30]
square: 5N+4N+3N 16 0.202...[30]
square: 6N+5N+2N 16 0.187...[30]
square: 6N+5N+3N 16 0.182...[30]
square: 6N+5N+4N 16 0.179...[30]
square: 6N+4N+3N+2N 16 0.208...[30]
square: 5N+4N+3N+2N 20 0.196...[30] 0.196 724(5)[32]
square: 6N+5N+3N+2N 20 0.177...[30]
square: 6N+5N+4N+2N 20 0.172...[30]
square: 6N+5N+4N+3N 20 0.167...[30]
square: 6N+5N+4N+3N+2N 24 0.164...[30]

2N = nearest neighbors, 3N = next-nearest neighbors, 4N = next-next-nearest neighbors, etc. These are called NN, 2NN, 3NN respectively in the 3D versions below.

For more neighbors, see [32]

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 = 1 [33]
honeycomb (63) 3
kagome (3, 6, 3, 6) 4 0.524430..., 3p2 + 6p3 - 12 p4+ 6 p5 - p6 = 1 [34]
(3, 4, 6, 4) 4
square (44) 4 1/2 (exact)
(34,6 ) 5 0.434371..., 12p3 + 36p4 -21 p5- 327 p6 + 69p7 + 2532p8 - 6533 p9

+ 8256 p10 - 6255p11 + 2951p12 - 837 p13+ 126 p14 - 7p15= 1 [35]

snub square, puzzle (32, 4, 3, 4 ) 5
(33, 42) 5
triangular (36) 6 1/2 (exact)

Formulas for site-bond percolation

Lattice z Threshold Notes
(63) honeycomb 3 3 ,

when equal: b = s = 0.82199

 approximate formula, s = site prob., b = bond prob., t = 1 - 2 sin (π/18) [13]

Archimedean Duals (Laves Lattices)

Laves lattices are the duals to the Archimedean lattices. Drawings from.[3] See also Uniform Tilings.

Lattice z Site Percolation Threshold Bond Percolation Threshold
Cairo pentagonal

D(32,4,3,4)=(2/3)(53)+(1/3)(54)

 3,4 3⅓ 0.6501834(2),[5] 0.650184(5)[3] 0.585863... = 1-pcbond(32,4,3,4)
Pentagonal D(33,42)=(1/3)(54)+(2/3)(53) 3,4 3⅓ 0.6470471(2),[5] 0.647084(5),[3] 0.6471(6) [28] 0.580358... = 1-pcbond(33,42), 0.5800(6) [28]
D(34,6)=(1/5)(46)+(4/5)(43) 3,6 3 3/5 0.639447[3] 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),[36] 0.585040(5)[3] 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)[3] 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,[37] 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)[38]

D(3, 122)=(2/3)(33)+(1/3)(312)

 3,12 6 1/2 0.259579... = 1-pcbond(3, 122)

Site bond percolation (both thresholds apply simultaneously to one system).

Lattice z Site Percolation Threshold Bond Percolation Threshold
square 4 4 0.615185(15)[39] 0.95
0.667280(15)[39] 0.85
0.732100(15)[39] 0.75
0.75 0.726195(15)[39]
0.815560(15)[39] 0.65
0.85 0.615810(30)[39]
0.95 0.533620(15)[39]

* For more values, see An Investigation of site-bond percolation

2-Uniform Lattices

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

[2]

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

[2]

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

[2]

Top 2 Lattices: #31 #32
Bottom Lattice: #33

[2]

# Lattice 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)[40] 0.67493252(36)[41]
42 (1/3)(3,4,6,4) + (2/3)(4,6,12) 4,3 3 1⁄3 0.7157(2)[40] 0.64536587(40)[41]
36 (1/7)(36) + (6/7)(32,4,12) 6,4 4  2⁄7 0.6808(2)[40] 0.55778329(40)[41]
15 (2/3)(32,62) + (1/3)(3,6,3,6) 4,4 4 0.6499(2)[40] 0.53632487(40)[41]
34 (1/7)(36) + (6/7)(32,62) 6,4 4  2⁄7 0.6329(2)[40] 0.51707873(70)[41]
16 (4/5)(3,42,6) + (1/5)(3,6,3,6) 4,4 4 0.6286(2)[40] 0.51891529(35)[41]
17 (4/5)(3,42,6) + (1/5)(3,6,3,6)* 4,4 4 0.6279(2)[40] 0.51769462(35)[41]
35 (2/3)(3,42,6) + (1/3)(3,4,6,4) 4,4 4 0.6221(2)[40] 0.51973831(40)[41]
11 (1/2)(34,6) + (1/2)(32,62) 5,4 4.5 0.6171(2)[40] 0.48921280(37)[41]
37 (1/2)(33,42) + (1/2)(3,4,6,4) 5,4 4.5 0.5885(2)[40] 0.47229486(38)[41]
30 (1/2)(32,4,3,4) + (1/2)(3,4,6,4) 5,4 4.5 0.5883(2)[40] 0.46573078(72)[41]
23 (1/2)(33,42) + (1/2)(44) 5,4 4.5 0.5720(2)[40] 0.45844622(40)[41]
22 (2/3)(33,42) + (1/3)(44) 5,4 4  2⁄3 0.5648(2)[40] 0.44528611(40)[41]
12 (1/4)(36) + (3/4)(34,6) 6,5 5  1⁄4 0.5607(2) [40] 0.41109890(37) [41]
33 (1/2)(33,42) + (1/2)(32,4,3,4) 5,5 5 0.5505(2) [40] 0.41628021(35) [41]
32 (1/3)(33,42) + (2/3)(32,4,3,4) 5,5 5 0.5504(2) [40] 0.41549285(36) [41]
31 (1/7)(36) + (6/7)(32,4,3,4) 6,5 5  1⁄7 0.5440(2) [40] 0.40379585(40) [41]
13 (1/2)(36) + (1/2)(34,6) 6,5 5.5 0.5407(2) [40] 0.38914898(35) [41]
21 (1/3)(36) + (2/3)(33,42) 6,5 5  1⁄3 0.5342(2) [40] 0.39491996(40) [41]
20 (1/2)(36) + (1/2)(33,42) 6,5 5.5 0.5258(2) [40] 0.38285085(38) [41]

Inhomogeneous 2-Uniform Lattice

This figure shows the 2-uniform lattice #37 in the isoradial representation in which each polygon is inscribed in a circle of unit radius. The squares in the 2-uniform lattice must now be represented as rectangles 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 condition [42] 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 2x2, 1x1 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 Site Percolation Threshold Bond Percolation Threshold
martini (3/4)(3,92)+(1/4)(93) 3 3 0.764826..., 1 +p4 - 3p3=0[43] 0.707107... = 1/√2 [44]
bow-tie (c) 3,4 3 1/7 0.672929..., 1-2p3-2p4-2p5-7p6+18p7+11p8-35p9+21p10-4p11=0 [45]
bow-tie (d) 3,4 3⅓ 0.625457..., 1-2p2-3p3+4p4-p5=0 [45]
martini-A (2/3)(3,72)+(1/3)(3,73) 3,4 3⅓ 1/√2[45] 0.625457..., 1-2p2-3p3+4p4-p5=0 [45]
bow-tie dual lattice (e) 3,4 3⅔ 0.595482..., 1-pcbond (bow-tie (a)) [45]
bow-tie (b) 3,4,6 3⅔ 0.533213..., 1-p- 2p3 -4p4-4p5+156+ 13p7-36p8+19p9+ p10 + p11=0 [45]
martini covering/medial (1/2)(33,9)+(1/2)(3,9,3,9) 4 4 0.707107... = 1/√2 [44] 0.57086651(33) [46]
martini-B (1/2)(3,5,3,52)+(1/2)(3,52) 3, 5 4 0.618034... = 2/(1 +√5)..., 1- p2-p=0[43][45] 1/2 [44][45]
bow-tie dual lattice (f) 3,4,8 4 2/5 0.466787..., 1-pcbond (bow-tie (b))[45]
bow-tie (a) (1/2)(32,4,32,4)+(1/2)(3,4,3) 4,6 5 0.5472(2) [28] 0.404518..., 1 - p - 6p2 +6p3-p5=0 [45]
bow-tie dual lattice (h) 3,6,8 5 0.374543..., 1-pcbond(bow-tie (d))[45]
bow-tie dual lattice (g) 3,6,10 5½ 0.547... = pcsite(bow-tie(a)) 0.327071..., 1-pcbond(bow-tie (c))[45]

Thresholds on 2D covering, medial, and matching lattices

Lattice z Site Percolation Threshold Bond Percolation Threshold
(4, 6, 12) covering/medial 4 4 pcbond(4, 6, 12) = 0.693731... 0.5593140(2),[5] 0.559315(1)[47]
(4, 82) covering/medial, square kagome 4 4 pcbond(4,82) = 0.676803... 0.544798017(4),[5] 0.54479793(34)[47]
(34, 6) medial 4 4 0.5247495(5)[5]
(3,4,6,4) medial 4 4 0.51276 [5]
(32, 4, 3, 4) medial 4 4 0.512682929(8)[5]
(33, 42) medial 4 4 0.5125245984(9)[5]
square covering (non-planar) 6 6 1/2 0.3371(1)[48]
square matching lattice (non-planar) 8 8 1 - pcsite(square) = 0.407253... 0.25036834(6)[12]

(4, 6, 12) covering/medial lattice

(4, 82) covering/medial lattice

(3,122) covering/medial lattice (in light grey), equivalent to the kagome (2 x 2) subnet, and in black, the dual of these lattices.

(left) (3,4,6,4) covering/medial lattice, (right) (3,4,6,4) medial dual, shown in red, with medial lattice in light gray behind it

Thresholds on 2D chimera non-planar lattices

Lattice z Site Percolation Threshold Bond Percolation Threshold
K(2,2) 4 4 0.51253(14)[49] 0.44778(15) [49]
K(3,3) 6 6 0.43760(15) [49] 0.35502(15) [49]
K(4,4) 8 8 0.38675(7) [49] 0.29427(12) [49]
K(5,5) 10 10 0.35115(13) [49] 0.25159(13) [49]
K(6,6) 12 12 0.32232(13)[49] 0.21942(11) [49]
K(7,7) 14 14 0.30052(14) [49] 0.19475(9) [49]
K(8,8) 16 16 0.28103(11) [49] 0.17496(10) [49]

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 [50]

Lattice z Site Percolation Threshold Bond Percolation Threshold
checkerboard – 2 × 2 subnet 4,3 0.596303(1) [51]
checkerboard – 4 × 4 subnet 4,3 0.633685(9) [51]
checkerboard – 8 × 8 subnet 4,3 0.642318(5) [51]
checkerboard – 16 × 16 subnet 4,3 0.64237(1) [51]
checkerboard- 32 × 32 subnet 4,3 0.64219(2) [51]
checkerboard – subnet 4,3 0.642216(10) [51]
kagome – 2 × 2 subnet = (3, 122) covering/medial 4 pcbond (3, 122) = 0.74042077... 0.600861966960(2),[5] 0.6008624(10),[13] 0.60086193(3)[9]
kagome – 3 × 3 subnet 4 0.6193296(10),[13] 0.61933176(5),[9] 0.61933044(32)[52]
kagome – 4 × 4 subnet 4 0.625365(3),[13] 0.62536424(7)[9]
kagome – subnet 4 0.628961(2) [13]
kagome – (1 × 1):(2 × 2) subnet = martini covering/medial 4 pcbond(martini) = 1/√2 = 0.707107... 0.57086648(36) [46]
kagome – (1 × 1):(3 × 3) subnet 4,3 0.728355596425196...[9] 0.58609776(37) [52]
kagome – (1 × 1):(4 × 4) subnet 0.738348473943256...[9]
kagome – (1 × 1):(5 × 5) subnet 0.743548682503071...[9]
kagome – (1 × 1):(6 × 6) subnet 0.746418147634282...[9]
kagome – (2 × 2):(3 × 3) subnet 0.61091770(30) [52]
triangular – 2 × 2 subnet 6,4 0.471628788 [51]
triangular – 3 × 3 subnet 6,4 0.509077793 [51]
triangular – 4 × 4 subnet 6,4 0.524364822 [51]
triangular – 5 × 5 subnet 6,4 0.5315976(10) [51]
triangular – subnet 6,4 0.53993(1) [51]

Thresholds of random sequentially adsorbed objects

system z Site Threshold
dimers on a square lattice 4 0.5617,[53] 0.562 [54]
dimers on a triangular lattice 6 0.4872(8)[55]
dimers and 5% impurities, triangular lattice 6 0.4832(7) [56]
linear 3-mers on a square lattice 4 0.528[54]
3-site 120° angle, 5% impurities, triangular lattice 6 0.4574(9)[56]
3-site triangles, 5% impurities, triangular lattice 6 0.5222(9)[56]
linear trimers and 5% impurities, triangular lattice 6 0.4603(8) [56]
linear 4-mers on a square lattice 4 0.504[54]
linear 5-mers on a square lattice 4 0.490[54]
linear 6-mers on a square lattice 4 0.479[54]
linear 8-mers on a square lattice 4 0.474[54]
linear 10-mers on a square lattice 4 0.469[54]
parallel dimers on a square lattice 4 0.5683[53]

The threshold gives the fraction of sites occupied by the objects when site percolation first takes place (not at full jamming). For longer dimers see Ref.[57]

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...[58]
Shifted covering, square lattice 6 0.347296...[58]
Staggered covering, square lattice 6 0.376825(2) [58]
Random covering, square lattice 6 0.367713(2) [58]
Parallel covering, triangular lattice 10 0.237418...[58]
Staggered covering, triangular lattice 10 0.237497(2) [58]
Random covering, triangular lattice 10 0.235340(1) [58]

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. [59]

l (polymer length) z Bond Percolation
1 4 0.5(exact) [60]
2 4 0.47697(4)[60]
4 4 0.44892(6) [60]
8 4 0.41880(4)[60]

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

k z Site Thresholds Bond Thresholds
1 4 0.593(2) [61] 0.5009(2) [61]
2 4 0.564(2) [61] 0.4859(2) [61]
3 4 0.552(2) [61] 0.4732(2) [61]
4 4 0.542(2) [61] 0.4630(2) [61]
5 4 0.531(2) [61] 0.4565(2) [61]
6 4 0.522(2) [61] 0.4497(2) [61]
7 4 0.511(2) [61] 0.4423(2) [61]
8 4 0.502(2) [61] 0.4348(2) [61]
9 4 0.493(2) [61] 0.4291(2) [61]
10 4 0.488(2) [61] 0.4232(2) [61]
11 4 0.482(2) [61] 0.4159(2) [61]
12 4 0.476(2) [61] 0.4114(2) [61]
13 4 0.471(2) [61] 0.4061(2) [61]
14 4 0.467(2) [61] 0.4011(2) [61]
15 4 0.4011(2) [61] 0.3979(2) [61]

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),[62] [33]

Thresholds for 2D continuum models

2D continuum percolation with disks
2D continuum percolation with ellipses of aspect ratio 2
System Φc ηc nc
Disks of radius r 0.6763475(6),[63] 0.676339(4) [64] 1.12808737(6),[65] 1.128085(2),[63] 1.128059(12) [64] 1.436322(2),[63] 1.436289(16) [64]
Ellipses, aspect ratio ε = 2 0.63 [66] 0.76 1.94
Ellipses, ε = 5 0.455 [67] 0.607 3.864
Ellipses, ε = 10 0.301 [67] 0.358 4.56
Ellipses, ε = 20 0.178 [67] 0.196 4.99
Ellipses, ε = 50 0.081 [67] 0.084 5.38
Ellipses, ε = 100 0.0417 [67] 0.0426 5.42
Ellipses, ε = 1000 0.0043 [67] 0.00431 5.5
Aligned squares of side 0.66674349(3),[65] 0.66653(1),[68] 0.6666(4)[69] 1.09884280(9),[65] 1.0982(3),[68] 1.098(1)[69] 1.09884280(9),[65] 1.0982(3),[68] 1.098(1)[69]
Randomly oriented squares 0.62554075(4),[65] 0.6254(2)[69] 0.9822723(1),[65] 0.9819(6)[69] 0.982278(14) [70] 0.9822723(1),[65] 0.9819(6)[69] 0.982278(14) [70]
Rectangles, ε = 1.1 0.624870(7) 0.980484(19) 1.078532(21) [70]
Rectangles, ε = 2 0.590635(5) 0.893147(13) 1.786294(26) [70]
Rectangles, ε = 3 0.5405983(34) 0.777830(7) 2.333491(22) [70]
Rectangles, ε = 4 0.4948145(38) 0.682830(8) 2.731318(30) [70]
Rectangles, ε = 5 0.4551398(31) 0.607226(6) 3.036130(28) [70]
Rectangles, ε = 10 0.3233507(25) 0.3906022(37) 3.906022(37) [70]
Rectangles, ε = 20 0.2048518(22) 0.2292268(27) 4.584535(54) [70]
Rectangles, ε = 50 0.09785513(36) 0.1029802(4) 5.149008(20) [70]
Rectangles, ε = 100 0.0523676(6) 0.0537886(6) 5.378856(60) [70]
Rectangles, ε = 200 0.02714526(34) 0.02752050(35) 5.504099(69) [70]
Rectangles, ε = 1000 0.00559424(6) 0.00560995(6) 5.609947(60) [70]
Sticks of length 5.6372858(6),[65] 5.63726(2) [71]
Power-law disks, x=2.05 0.993(1) [72] 4.90(1) 0.0380(6)
Power-law disks, x=2.25 0.8591(5) [72] 1.959(5) 0.06930(12)
Power-law disks, x=2.5 0.7836(4) [72] 1.5307(17) 0.09745(11)
Power-law disks, x=4 0.69543(6) [72] 1.18853(19) 0.18916(3)
Power-law disks, x=5 0.68643(13) [72] 1.1597(3) 0.22149(8)
Power-law disks, x=6 0.68241(8) [72] 1.1470(1) 0.24340(5)
Power-law disks, x=7 0.6803(8) [72] 1.140(6) 0.25933(16)
Power-law disks, x=8 0.67917(9) [72] 1.1368(5) 0.27140(7)
Power-law disks, x=9 0.67856(12) [72] 1.1349(4) 0.28098(9)
Voids around disks of radius r 1 - Φc(disk) = 0.32355169(2),[65] 0.318(2),[73] 0.3261(6)[74]

equals critical total area for disks, where N is the number of objects and L is the system size.

for ellipses of semi-major and semi-minor axes of a and b, respectively. Aspect ratio with .

for rectangles of dimensions and . Aspect ratio with .

for power-law distributed disks with , .

equals critical area fraction.

equals number of objects of maximum length per unit area.

For ellipses,

For void percolation, is the critical void fraction.

For more ellipse values, see [66]

For more rectangle values, see [70]

Thresholds on 2D random and quasi-lattices

Voronoi diagram (solid lines) and its dual, the Delaunay triangulation (dotted lines), for a Poisson distribution of points
Delaunay triangulation
The Voronoi covering or line graph (dotted red lines) and the Voronoi diagram (black lines)
The Relative Neighborhood Graph (black lines) [75] superimposed on the Delaunay triangulation (black plus grey lines).
The Gabriel Graph, a subgraph of the Delaunay triangulation in which the circle surrounding each edge does not enclose any other points of the graph
Lattice z Site Percolation Threshold Bond Percolation Threshold
Relative neighborhood graph 2.5576 0.796(2) [75] 0.771(2) [75]
Voronoi tessellation 3 0.71410(2),[76] 0.7151* [40] 0.68,[77] 0.666931(5),[76] 0.6670(1) [78]
Voronoi covering/medial 4 0.666931(2)[76][78] 0.53618(2) [76]
Randomized kagome/square-octagon, fraction r=1/2 4 0.6599[10]
Penrose rhomb dual 4 0.6381(3)[36] 0.5233(2) [36]
Gabriel graph 4 0.6348(8),[79] 0.62[80] 0.5167(6),[79] 0.52[80]
Random-line tessellation, dual 4 0.586(2)[81]
Penrose rhomb 4 0.5837(3),[36] 0.58391(1)[82] 0.4770(2) [36]
Octagonal lattice, "chemical" links (Ammann Beenker tiling) 4 0.585 [83] 0.48 [83]
Octagonal lattice, "ferromagnetic" links 5.17 0.543 [83] 0.40 [83]
Dodecagonal lattice, "chemical" links 3.63 0.628 [83] 0.54 [83]
Dodecagonal lattice, "ferromagnetic" links 4.27 0.617 [83] 0.495 [83]
Delaunay triangulation 6 1/2 [84] 0.333069(2) [76][78]

*Theoretical estimate

Thresholds on slabs

Lattice z Site Percolation Threshold 0.67634831(2),[65] Bond Percolation Threshold
h= 2, SC, open b.c. 0.47424 [85]
h = 3, BCC, periodic b.c. 0.21113018(38) [86]
h = 4, BCC, periodic b.c. 0.20235168(59) [86]
h= 4, SC, open b.c. 0.3997 [85]
h = 5, SC, periodic b.c. 0.278102(5) [86]
h = 6, SC, periodic b.c. 0.272380(2) [86]
h = 7, SC, periodic b.c. 5,6 5,6 0.3459514(12) [86] 0.268459(1) [86]
h= 8, SC, open b.c. 0.3557 [85]
h = 8, SC, periodic b.c. 0.265615(5) [86]

More for SC open b.c. in Ref.[85]

h is the thickness of the slab, h x ∞ x ∞.

Thresholds on 3D lattices

Lattice z filling factor* filling fraction* Site Percolation Threshold Bond Percolation Threshold
(10,3)-a oxide (or site-bond) [87] 2.4 0.748713(22)[87]
(10,3)-b oxide (or site-bond) [87] 2.4 0.745317(25)[87]
silicon dioxide (diamond site-bond) [87] 2 ⅔ 0.638683(35)[87]
(8,3)-a[88] 3 3 0.577962(33)[88] 0.555700(22)[88]
(10,3)-a[88] 3 3 0.571404(40)[88] 0.551060(37)[88]
(10,3)-b[88] 3 3 0.565442(40)[88] 0.546694(33)[88]
cubic oxide (cubic site-bond)[87] 3 0.524652(50)[87]
bcc dual 4 0.4560(6) [89] 0.4031(6) [89]
ice Ih 4 4 π √3 / 16 = 0.340087 0.147 0.433(11)[90] 0.388(10)[91]
diamond (Ice Ic) 4 4 π √3 / 16 = 0.340087 0.1462332 0.4299870(4),[92] 0.426(+0.08,-0.02),[93]

0.4301(4),[94] 0.428(4),[95] 0.425(15),[96] 0.436(12),[90]

 0.3895892(5),[92] 0.3893(2),[94]

0.388(5),[96] 0.388(5) [95] 0.390(11),[91]

diamond dual 6 2/3 0.3904(5) [89] 0.2350(5)[89]
3D kagome (covering graph of the diamond lattice) 6 π √2 / 12 = 0.37024 0.1442 0.3895(2)[97] =pc(site) for diamond dual and pc(bond) for diamond lattice [89] 0.2709(6)[89]
Bow-tie stack dual 5⅓ 0.3480(4)[28] 0.2853(4)[28]
honeycomb stack 5 5 0.3701(2) [28] 0.3093(2) [28]
octagonal stack dual 5 5 0.3840(4) [28] 0.3168(4) [28]
pentagonal stack 5⅓ 0.3394(4) [28] 0.2793(4) [28]
kagome stack 6 6 0.453450 0.1517 0.3346(4) [28] 0.2563(2) [28]
fcc dual 42,8 5 1/3 0.3341(5) [89] 0.2703(3) [89]
simple cubic 6 6 π / 6 = 0.5235988 0.1631574 0.307(10) [96] 0.3115(5),[98] 0.3116077(2),[99] 0.311604(6),[100]

0.311605(5),[101] 0.311600(5),[102] 0.3116077(4),[103] 0.3116081(13),[104] 0.3116080(4),[105] 0.3116004(35),[106] 0.31160768(15)[92]

 0.247(5) [96]

0.24881182(10),[99] 0.2488125(25),[107] 0.2488126(5) [108]

hcp dual 44,82 5 1/3 0.3101(5) [89] 0.2573(3) [89]
dice stack 5,8 6 π √3 / 9 = 0.604600 0.1813 0.2998(4) [28] 0.2378(4) [28]
bow-tie stack 7 7 0.2822(6) [28] 0.2092(4) [28]
Stacked triangular / simple hexagonal 8 8 0.26240(5),[109] 0.2625(2),[110] 0.2623(2)[28] 0.18602(2),[109] 0.1859(2) [28]
octagonal (union-jack) stack 6,10 8 0.2524(6) [28] 0.1752(2) [28]
bcc 8 8 0.243(10) [96]

0.2459615(10),[105] 0.2460(3),[111] 0.2464(7),[112] 0.2458(2)[94]

 0.178(5) [96]

0.1802875(10)[108]

simple cubic with 3NN 8 8 0.2455(1) [113]
simple cubic with 3NN+4NN 14 14 0.20490(12) [114]
fcc 12 12 π / (3 √2) = 0.740480 0.147530 0.198(3),[115] 0.1992365(10),[105] 0.19923517(20),[92] 0.1994(2)[94] 0.1201635(10)[108]
hcp 12 12 π / (3 √2) = 0.740480 0.147545 0.195(5)[96]

0.1992555(10)[116]

 0.1201640(10)[116]

0.119(2)[96]

La2-x Srx Cu O4 12 12 0.19927(2) [117]
simple cubic with 2NN 12 12 0.1991(1) [113]
BCC+2NN 14 14 0.1686(20)[118] 0.0991(5)[118]
simple cubic with 2NN+4NN 18 18 0.15950(12)[114]
simple cubic with NN+4NN 12 12 0.15040(12) [114]
simple cubic with NN+3NN 14 14 0.1420(1) [113]
simple cubic with NN+2NN 18 18 0.1372(1),[113] 0.13735(5) [119]
simple cubic with short-length correlation 6+ 6+ 0.126(1)[120]
simple cubic with NN+3NN+4NN 20 20 0.11920(12) [114]
simple cubic with NN+2NN+4NN 24 24 0.11440(12) [114]
simple cubic with 2NN+3NN+4NN 26 26 0.11330(12) [114]
simple cubic with 2NN+3NN 20 20 0.1036(1) [113]
simple cubic with NN+2NN+3NN+4NN 32 32 0.10000(12) [114]
simple cubic with NN+2NN+3NN 26 26 0.0976(1),[113] 0.0976445(10) [119]

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

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

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

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 [121]

System polymer Φc
percolating excluded volume of athermal polymer matrix (bond-fluctuation model on cubic lattice) 0.4304(3) [122]

Dimer percolation in 3D

System Site Percolation Threshold Bond Percolation Threshold
Simple cubic 0.2555(1)[123]

Thresholds for 3D continuum models

All overlapping except for jammed spheres and polymer matrix.

System Φc ηc
Spheres of radius r 0.2895(5),[124] 0.289573(2),[125] 0.2854 [126] 0.3418(7),[124] 0.341889(3),[125] 0.3360,[126]

0.34189(2),[68] [corrected]

Oblate ellipsoids with major radius r and aspect ratio 4/3 0.2831 [126] 0.3328[126]
Prolate ellipsoids with minor radius r and aspect ratio 3/2 0.2795 [126] 0.3278[126]
Oblate ellipsoids with major radius r and aspect ratio 2 0.2629 [126] 0.3050[126]
Prolate ellipsoids with minor radius r and aspect ratio 2 0.2618,[126] 0.25(2)[127] 0.3035,[126] 0.29(3) [127]
Oblate ellipsoids with major radius r and aspect ratio 3 0.2289 [126] 0.2599[126]
Prolate ellipsoids with minor radius r and aspect ratio 3 0.2244,[126] 0.20(2)[127] 0.2541,[126] 0.22(3)[127]
Oblate ellipsoids with major radius r and aspect ratio 4 0.2003 [126] 0.2235[126]
Prolate ellipsoids with minor radius r and aspect ratio 4 0.1901,[126] 0.16(2)[127] 0.2108,[126] 0.17(3)[127]
Oblate ellipsoids with major radius r and aspect ratio 5 0.1757 [126] 0.1932[126]
Prolate ellipsoids with minor radius r and aspect ratio 5 0.1627,[126] 0.13(2)[127] 0.1776,[126] 0.15(2)[127]
Oblate ellipsoids with major radius r and aspect ratio 10 0.1058 [126] 0.1118[126]
Prolate ellipsoids with minor radius r and aspect ratio 10 0.08703,[126] 0.07(2)[127] 0.09105,[126] 0.07(2)[127]
Oblate ellipsoids with major radius r and aspect ratio 100 0.01248[126] 0.01256[126]
Prolate ellipsoids with minor radius r and aspect ratio 100 0.006949[126] 0.006973[126]
Oblate ellipsoids with major radius r and aspect ratio 1000 0.001275 [126] 0.001276 [126]
Oblate ellipsoids with major radius r and aspect ratio 2000 0.000637[126] 0.000637 [126]
Aligned cylinders 0.2819(2)[128] 0.3312(1)[128]
Aligned cubes of side 0.2773(2) [69] 0.27727(2)[32] 0.3247(3),[68] 0.3248(3)[69]
Randomly oriented icosahedra 0.3030(5) [129]
Randomly oriented dodecahedra 0.2949(5) [129]
Randomly oriented octahedra 0.2514(6) [129]
Randomly oriented cubes of side 0.2168(2) [69] 0.2444(3),[69] 0.2443(5)[129]
Randomly oriented tetrahedra 0.1701(7) [129]
Randomly oriented disks of radius r (in 3D) 0.9614(5)[130]
Randomly oriented square plates of side 0.8647(6)[130]
Randomly oriented triangular plates of side 0.7295(6)[130]
Voids around disks of radius r 22.86(2)[131]
Voids around oblate ellipsoids of major radius r and aspect ratio 10 15.42(1)[131]
Voids around oblate ellipsoids of major radius r and aspect ratio 2 6.478(8)[131]
Voids around spheres of radius r 0.034(7),[132] 0.030(2),[73] 0.0301(3),[133] 0.0294,[134] 0.0300(3) [135] 0.0317(4) [136] 3.506(8),[135] 3.515(6) [131]
Voids around aligned cubes 0.036(1) [32]
Jammed spheres (average z = 6) 0.183(3),[137] 0.1990 [138] 0.59(1) [137]

is the total volume, where N is the number of objects and L is the system size.

is the critical volume fraction.

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

For void ("Swiss-Cheese" model), is the critical void fraction.

For more results on void percolation around ellipsoids and elliptical plates, see.[131]

For more ellipsoid percolation values see [126]

Thresholds on 3D random and quasi-lattices

Lattice z Site Percolation Threshold Bond Percolation Threshold
Contact network of packed spheres 6 0.310(5) [137]
Random-plane tessellation, dual 6 0.290(7) [139]
Icosahedral Penrose 6 0.285[140] 0.225 [140]
Penrose w/2 diagonals 6.764 0.271[140] 0.207 [140]
Penrose w/8 diagonals 12.764 0.188[140] 0.111[140]
Voronoi network 15.54 0.1453(20) [118] 0.0822(50)[118]

Thresholds for 3D correlated percolation

Lattice z Site Percolation Threshold Bond Percolation Threshold
Drilling percolation, simple cubic lattice 6 6 *0.633965(15),[141] 0.6339(5)

,[142] 6345(3) [143]

Continuum models in higher dimensions

d System Φc ηc
4 Overlapping hyperspheres 0.1223(4)[68] 0.1304(5) [68]
4 Overlapping hypercubes 0.1132(5)[68] 0.1201(6) [68]
4 Voids around hyperspheres 0.00211(2)[74] 6.161(10)[74]
5 Overlapping hyperspheres 0.05443(7) [68]
5 Overlapping hypercubes 0.05024(7) [68]
5 Voids around hyperspheres 1.26(6)x10−4 [74] 8.98(4)[74]
6 Overlapping hyperspheres 0.02339(5) [68]
6 Overlapping hypercubes 0.02104(8)[68]
6 Voids around hyperspheres 8.0(6)x10−6 [74] 11.74(8)[74]

.

In 4d, .

In 5d, .

In 6d, .

is the critical volume fraction.

For void models, is the critical void fraction, and is the total volume of the overlapping objects

Thresholds on hypercubic lattices

d z Site Thresholds Bond Thresholds
4 8 0.1968861(14),[144] 0.196889(3),[145] 0.196901(5) [146] 0.1601314(13),[144] 0.160130(3),[145] 0.1601310(10) [107]
5 10 0.1407966(15) [144] 0.118172(1),[144] 0.1181718(3) [107]
6 12 0.109017(2) [144] 0.0942019(6) [144]
7 14 0.0889511(9),[144] 0.088939(20) [147] 0.0786752(3) [144]
8 16 0.0752101(5) [144] 0.06770839(7) [144]
9 18 0.0652095(3) [144] 0.05949601(5) [144]
10 20 0.0575930(1) [144] 0.05309258(4) [144]
11 22 0.05158971(8) [144] 0.04794969(1) [144]
12 24 0.04673099(6) [144] 0.04372386(1) [144]
13 26 0.04271508(8) [144] 0.04018762(1) [144]

Thresholds in higher-dimensional lattices

d lattice z Site Thresholds Bond Thresholds
4 diamond 5 0.2978(2)[94]
4 kagome 8 0.2715(3) [97]
4 fcc 24 0.0842(3)[94] 0.049(1)[94]
5 diamond 6 0.2252(3)[94]
5 kagome 10 0.2084(4) [97]
5 bcc 32 0.0446(4)[94] 0.033(1)[94]
5 fcc 40 0.0431(3)[94] 0.026(2)[94]
6 diamond 7 0.1799(5)[94]
6 kagome 12 0.1677(7) [97]
6 fcc 60 0.0252(5)[94]
6 bcc 64 0.0199(5)[94]

Thresholds on hyperbolic, hierarchical, and tree lattices

Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk [148]
Depiction of the non-planar Hanoi network HN-NP [149]
Lattice z Site Percolation Threshold Bond Percolation Threshold
Lower Upper
{4,5} hyperbolic 5 5 0.27[150] 0.52[150]
{7,3} hyperbolic 3 3 0.72[150] 0.53[150]
{3,7} hyperbolic 7 7 0.20[150] 0.37[150]
{∞,3} Cayley tree 3 3 1/2 1/2[150] 1[150]
Enhanced binary tree (EBT) 0.304(1)[150] 0.48,[150] 0.564(1)[151]
Enhanced binary tree dual 0.436(1)[151] 0.696(1)[151]
Non-Planar Hanoi Network (HN-NP) 0.319445[149] 0.381996[149]
Cayley tree with grandparents 8 0.158656326[152]

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

Cayley tree (Bethe lattice) with coordination number z: pc= 1 / (z - 1)

Cayley tree with a distribution of z with mean , mean-square pc= [153] (site or bond threshold)

Thresholds for directed percolation

(1+1)D Kagome Lattice
(1+1)D Square Lattice
(1+1)D Triangular Lattice
(2+1)D SC Lattice
(2+1)D BCC Lattice
Lattice z Site Percolation Threshold Bond Percolation Threshold
(1+1)-d honeycomb 1.5 0.8399316(2),[154] 0.839933(5),[155] 0.8228569(2),[154] 0.82285680(6)[154]
(1+1)-d kagome 2 0.7369317(2),[154] 0.73693182(4)[156] 0.6589689(2),[154] 0.65896910(8)[154]
(1+1)-d square, diagonal direction 2 0.705489(4),[157] 0.70548522(4),[158] 0.70548515(20),[156]

0.7054852(3),[154]

 0.644701(2),[159] 0.644701(1),[160] 0.64470015(5),[161] 0.644700185(5),[158] 0.6447001(2),[154]
(1+1)-d triangular 3 0.5956468(5),[161] 0.5956470(3) [154] 0.478025(1),[161] 0.4780250(4) [154]
(2+1)-d simple cubic, diagonal planes 3 0.43531(1) [162] 0.382223(7) [162]
(2+1)-d square nn (= bcc) 4 0.3445736(3),[163] 0.344575(15) [164] 0.2873383(1),[165] 0.287338(3)[162]
(3+1)-d hypercubic, diagonal planes 4 0.3025(10) [166]
(3+1)-d cubic, nn 6 0.2081040(4) [163] 0.1774970(5) [107]
(3+1)-d body-centered hypercubic 8 0.160950(30) [164]
(4+1)-d hypercubic, nn 8 0.1461593(2),[163] 0.1461582(3) [167] 0.1288557(5) [107]
(4+1)-d body-centered hypercubic 16 0.075582(17) [164]

0.0755850(3) [167]
(5+1)-d hypercubic, nn 10 0.1123373(2) [163] 0.1016796(5) [107]
(5+1)-d body-centered hypercubic 32 0.035967(23) [164]
(6+1)-d hypercubic, nn 12 0.0913087(2) [163] 0.0841997(14) [107]
(7+1)-d hypercubic,nn 14 0.07699336(7) [163] 0.07195(5) [107]

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.

Exact critical manifolds of inhomogeneous systems

Inhomogeneous triangular lattice bond percolation[14]

Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation[14]

Inhomogeneous (3,12^2) lattice, site percolation[4] [168]

or

Inhomogeneous union-jack lattice, site percolation with probabilities [169]

Inhomogeneous martini lattice, bond percolation [45][170]

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

Inhomogeneous martini-A (3–7) lattice, bond percolation. Left side (top of "A" to bottom): . Right side: . Cross bond: .

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

Inhomogeneous martini lattice with outside enclosing triangle of bonds, probabilities from inside to outside, bond percolation [170]

Inhomogeneous checkerboard lattice, bond percolation [34][62]

Inhomogeneous bow-tie lattice, bond percolation [33][62]

where are the four bonds around the square and is the diagonal bond connecting the vertex between bonds and .

Percolation thresholds for graphs

For random graphs not embedded in space the percolation threshold can be calculated exactly. For example, for random regular graphs where all nodes have the same degree k, pc=1/k. For Erdős–Rényi (ER) graphs with Poissonian degree distribution, pc=1/<k>.[171] The critical threshold was calculated exactly also for a network of interdependent ER networks.[172][173]

See also

References

  1. ↑ Kasteleyn, P. W.; C. M. Fortuin (1969). "Phase transitions in lattice systems with random local properties". Journal of the Physical Society of Japan (Supplements). 26: 11–14.
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