Percolation threshold

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Percolation threshold is a mathematical term 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[edit]

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 Shepard,[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[edit]

Example image caption


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 \overline 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]
honeycomb (63) 3 3 0.6962(6),[10] 0.697040230(5),[5] 0.6970402(1),[11] 0.6970413(10),[12] 0.697043(3),[4] 0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=0[13]
kagome (3, 6, 3, 6) 4 4 0.652703645... = 1 - 2 sin(π/18)[13] 0.524404978(5),[9] 0.52440499(2),[11]

0.52440572...,[14] 0.52440500(1),[7] 0.52440516(10),[12] 0.5244053(3),[15] 0.524404999173(3),[5][6]

ruby,[16] 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 0.59274601(2),[5] 0.59274621(13),[17] 0.59274621(33),[18] 0.59274598(4),[19][20] 0.59274605(3),[11] 0.593(1)[21] 1/2
snub hexagonal, maple leaf [22] (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]
(33, 42) 5 5 0.550213(3),[4] 0.5502(8)[23] 0.41964044(1),[5] 0.41964191(43),[8] 0.4196(6) [23]
triangular (36) 6 6 1/2 0.347296355... = 2 sin (π/18), 1+ p3-3p=0[13]

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[edit]

Lattice z Site Percolation Threshold Bond Percolation Threshold
square: 3N, 4N, 6N 4 0.592...[24][25]
square: 3N+2N, 4N+3N, 6N+4N 8 0.407...[24][25][26]
square: 4N+2N 8 0.337...[24][25]
square: 6N+3N 8 0.337...[25]
square: 5N 8 0.270...[25]
square: 6N+2N 8 0.277...[25]
square: 4N+3N+2N 12 0.288...[24][25]
square: 6N+4N+3N 12 0.288...[25]
square: 5N+2N 12 0.236...[25]
square: 5N+3N 12 0.225...[25]
square: 5N+4N 12 0.221...[25]
square: 6N+3N+2N 12 0.240...[25]
square: 6N+4N+2N 12 0.233...[25]
square: 6N+5N 12 0.199...[25]
square: 5N+3N+2N 16 0.219...[25]
square: 5N+4N+2N 16 0.208...[25]
square: 5N+4N+3N 16 0.202...[25]
square: 6N+5N+2N 16 0.187...[25]
square: 6N+5N+3N 16 0.182...[25]
square: 6N+5N+4N 16 0.179...[25]
square: 6N+4N+3N+2N 16 0.208...[25]
square: 5N+4N+3N+2N 20 0.196...[25]
square: 6N+5N+3N+2N 20 0.177...[25]
square: 6N+5N+4N+2N 20 0.172...[25]
square: 6N+5N+4N+3N 20 0.167...[25]
square: 6N+5N+4N+3N+2N 24 0.164...[25]

2N = nearest neighbours, 3N = next-nearest neighbours, 4N = next-next-nearest neighbours, 5N = next-next-next-nearest neighbours, etc.

Approximate formulas for thresholds of Archimedean lattices[edit]

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 [27]
honeycomb (63) 3
kagome (3, 6, 3, 6) 4 0.524430..., 3p2 + 6p3 - 12 p4+ 6 p5 - p6 = 1 [28]
(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 [29]

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

Formulas for site-bond percolation[edit]

Lattice z  \overline z Threshold Notes
(63) honeycomb 3 3  b s [1 - (\sqrt{t}/(3-t))(\sqrt{b} - \sqrt{t})] = t ,

when equal: b = s = 0.82199

approximate formula, s = site prob., b = bond prob., t = 1 - 2 sin (π/18) [12]

Archimedean Duals (Laves Lattices)[edit]

Example image caption

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

Lattice z \overline 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)
D(33,42)=(1/3)(54)+(2/3)(53) 3,4 3⅓ 0.6470471(2),[5] 0.647084(5)[3] 0.580358... = 1-pcbond(33,42)
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),[30] 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,[31] 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)[32]

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 \overline z Site Percolation Threshold Bond Percolation Threshold
square 4 4 0.615185(15)[33] 0.95
0.667280(15)[33] 0.85
0.732100(15)[33] 0.75
0.75 0.726195(15)[33]
0.815560(15)[33] 0.65
0.85 0.615810(30)[33]
0.95 0.533620(15)[33]

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

2-Uniform Lattices[edit]

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

20 2 uniform lattices

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

20 2 uniform lattices

[2]
Top 4 Lattices: #22 #23 #21 #20
Bottom 3 Lattices: #16 #17 #15

20 2 uniform lattices

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

20 2 uniform lattices

[2]

# Lattice z \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)[34] 0.67493252(36)[35]
42 (1/3)(3,4,6,4) + (2/3)(4,6,12) 4,3 313 0.7157(2)[34] 0.64536587(40)[35]
36 (1/7)(36) + (6/7)(32,4,12) 6,4 4 27 0.6808(2)[34] 0.55778329(40)[35]
15 (2/3)(32,62) + (1/3)(3,6,3,6) 4,4 4 0.6499(2)[34] 0.53632487(40)[35]
34 (1/7)(36) + (6/7)(32,62) 6,4 4 27 0.6329(2)[34] 0.51707873(70)[35]
16 (4/5)(3,42,6) + (1/5)(3,6,3,6) 4,4 4 0.6286(2)[34] 0.51891529(35)[35]
17 (4/5)(3,42,6) + (1/5)(3,6,3,6)* 4,4 4 0.6279(2)[34] 0.51769462(35)[35]
35 (2/3)(3,42,6) + (1/3)(3,4,6,4) 4,4 4 0.6221(2)[34] 0.51973831(40)[35]
11 (1/2)(34,6) + (1/2)(32,62) 5,4 4.5 0.6171(2)[34] 0.48921280(37)[35]
37 (1/2)(33,42) + (1/2)(3,4,6,4) 5,4 4.5 0.5885(2)[34] 0.47229486(38)[35]
30 (1/2)(32,4,3,4) + (1/2)(3,4,6,4) 5,4 4.5 0.5883(2)[34] 0.46573078(72)[35]
23 (1/2)(33,42) + (1/2)(44) 5,4 4.5 0.5720(2)[34] 0.45844622(40)[35]
22 (2/3)(33,42) + (1/3)(44) 5,4 4 23 0.5648(2)[34] 0.44528611(40)[35]
12 (1/4)(36) + (3/4)(34,6) 6,5 5 14 0.5607(2) [34] 0.41109890(37) [35]
33 (1/2)(33,42) + (1/2)(32,4,3,4) 5,5 5 0.5505(2) [34] 0.41628021(35) [35]
32 (1/3)(33,42) + (2/3)(32,4,3,4) 5,5 5 0.5504(2) [34] 0.41549285(36) [35]
31 (1/7)(36) + (6/7)(32,4,3,4) 6,5 5 17 0.5440(2) [34] 0.40379585(40) [35]
13 (1/2)(36) + (1/2)(34,6) 6,5 5.5 0.5407(2) [34] 0.38914898(35) [35]
21 (1/3)(36) + (2/3)(33,42) 6,5 5 13 0.5342(2) [34] 0.39491996(40) [35]
20 (1/2)(36) + (1/2)(33,42) 6,5 5.5 0.5258(2) [34] 0.38285085(38) [35]

Inhomogeneous 2-Uniform Lattice[edit]

2uniformLattice37

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 [36] 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[edit]

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).

Example image caption


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

Example image caption
Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
martini (3/4)(3,92)+(1/4)(93) 3 3 0.764826..., 1 +p4 - 3p3=0[37] 0.707107... = 1/√2 [38]
bow-tie (c) 3,4 3 1/7 0.672929..., 1-2p3-2p4-2p5-7p6+18p7+11p8-35p9+21p10-4p11=0 [39]
bow-tie (d) 3,4 3⅓ 0.625457..., 1-2p2-3p3+4p4-p5=0 [39]
martini-A (2/3)(3,72)+(1/3)(3,73) 3,4 3⅓ 1/√2[39] 0.625457..., 1-2p2-3p3+4p4-p5=0 [39]
bow-tie dual lattice (e) 3,4 3⅔ 0.595482..., 1-pcbond (bow-tie (a)) [39]
bow-tie (b) 3,4,6 3⅔ 0.533213..., 1-p- 2p3 -4p4-4p5+156+ 13p7-36p8+19p9+ p10 + p11=0 [39]
martini covering/medial (1/2)(33,9)+(1/2)(3,9,3,9) 4 4 0.707107... = 1/√2 [38] 0.57086651(33) [40]
martini-B (1/2)(3,5,3,52)+(1/2)(3,52) 3, 5 4 0.618034... = 2/(1 +√5)..., 1- p2-p=0[37][39] 1/2 [38][39]
bow-tie dual lattice (f) 3,4,8 4 2/5 0.466787..., 1-pcbond (bow-tie (b))[39]
bow-tie (a) (1/2)(32,4,32,4)+(1/2)(3,4,3) 4,6 5 0.5472(2) [23] 0.404518..., 1 - p - 6p2 +6p3-p5=0 [39]
bow-tie dual lattice (h) 3,6,8 5 0.374543..., 1-pcbond(bow-tie (d))[39]
bow-tie dual lattice (g) 3,6,10 0.547... = pcsite(bow-tie(a)) 0.327071..., 1-pcbond(bow-tie (c))[39]

Thresholds on 2D covering, medial, and matching lattices[edit]

Lattice z \overline 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)[41]
(4, 82) covering/medial, square kagome 4 4 pcbond(4,82) = 0.676803... 0.544798017(4),[5] 0.54479793(34)[41]
(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)[42]
square matching lattice (non-planar) 8 8 1 - pcsite(square) = 0.407253... 0.25036834(6)[11]
4, 6, 12, Covering/medial lattice

(4, 6, 12) covering/medial lattice

(4, 8^2) Covering/medial lattice

(4, 82) covering/medial lattice

(3,12^2) 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.

(3,4,6,4) medial lattice
(3,4,6,4) medial dual

(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 subnet lattices[edit]

Example image caption

The 2 × 2 subnet is known as the "triangular kagome" lattice [43]

Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
checkerboard – 2 × 2 subnet 4,3 0.596303(1) [44]
checkerboard – 4 × 4 subnet 4,3 0.633685(9) [44]
checkerboard – 8 × 8 subnet 4,3 0.642318(5) [44]
checkerboard – 16 × 16 subnet 4,3 0.64237(1) [44]
checkerboard- 32 × 32 subnet 4,3 0.64219(2) [44]
checkerboard – \infty subnet 4,3 0.642216(10) [44]
kagome – 2 × 2 subnet = (3, 122) covering/medial 4 pcbond (3, 122) = 0.74042077... 0.600861966953(1), 0.6008624(10),[12] 0.60086193(3)[9]
kagome – 3 × 3 subnet 4 0.6193296(10),[12] 0.61933176(5),[9] 0.61933044(32)[45]
kagome – 4 × 4 subnet 4 0.625365(3),[12] 0.62536424(7)[9]
kagome – \infty subnet 4 0.628961(2) [12]
kagome – (1 × 1):(2 × 2) subnet = martini covering/medial 4 pcbond(martini) = 1/√2 = 0.707107... 0.57086648(36) [40]
kagome – (1 × 1):(3 × 3) subnet 4,3 0.728355596425196...[9] 0.58609776(37) [45]
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) [45]
triangular – 2 × 2 subnet 6,4 0.471628788 [44]
triangular – 3 × 3 subnet 6,4 0.509077793 [44]
triangular – 4 × 4 subnet 6,4 0.524364822 [44]
triangular – 5 × 5 subnet 6,4 0.5315976(10) [44]
triangular – \infty subnet 6,4 0.53993(1) [44]

Thresholds of dimers a square lattice[edit]

system z Site Threshold
unoriented dimers 4 0.5617 [46]
parallel dimers 4 0.5683[46]

Thresholds of polymers (random walks) on a square lattice[edit]

System is composed of ordinary (non-avoiding) random walks of length l on the square lattice. [47]

l (polymer length) z Bond Percolation
1 4 0.5(exact) [48]
2 4 0.47697(4)[48]
4 4 0.44892(6) [48]
8 4 0.41880(4)[48]

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

k z Site Thresholds Bond Thresholds
1 4 0.593(2) [49] 0.5009(2) [49]
2 4 0.564(2) [49] 0.4859(2) [49]
3 4 0.552(2) [49] 0.4732(2) [49]
4 4 0.542(2) [49] 0.4630(2) [49]
5 4 0.531(2) [49] 0.4565(2) [49]
6 4 0.522(2) [49] 0.4497(2) [49]
7 4 0.511(2) [49] 0.4423(2) [49]
8 4 0.502(2) [49] 0.4348(2) [49]
9 4 0.493(2) [49] 0.4291(2) [49]
10 4 0.488(2) [49] 0.4232(2) [49]
11 4 0.482(2) [49] 0.4159(2) [49]
12 4 0.476(2) [49] 0.4114(2) [49]
13 4 0.471(2) [49] 0.4061(2) [49]
14 4 0.467(2) [49] 0.4011(2) [49]
15 4 0.4011(2) [49] 0.3979(2) [49]

Thresholds on 2D inhomogeneous lattices[edit]

Lattice z Site Percolation Threshold Bond Percolation Threshold
bow-tie with p = 1/2 on one non-diagonal bond 3 0.3819654(5),[50] (3 - \sqrt{5})/2 [27]

Thresholds for 2D continuum models[edit]

2D continuum percolation with disks
2D continuum percolation with ellipses of aspect ratio 2
System Φc ηc nc
Disks of radius r 0.67634831(2),[51] 0.6763475(6),[52] 0.676339(4) [53] 1.12808737(6),[51] 1.128085(2),[52] 1.128059(12) [53] 1.436322(2),[52] 1.436289(16) [53]
Ellipses, aspect ratio ε = 2 0.63 [54] 0.76 1.94
Ellipses, ε = 5 0.455 [55] 0.607 3.864
Ellipses, ε = 10 0.301 [55] 0.358 4.56
Ellipses, ε = 20 0.178 [55] 0.196 4.99
Ellipses, ε = 50 0.081 [55] 0.084 5.38
Ellipses, ε = 100 0.0417 [55] 0.0426 5.42
Ellipses, ε = 1000 0.0043 [55] 0.00431 5.5
Aligned squares of side \ell 0.66674349(3),[51] 0.66653(1),[56] 0.6666(4)[57] 1.09884280(9),[51] 1.0982(3),[56] 1.098(1)[57] 1.09884280(9),[51] 1.0982(3),[56] 1.098(1)[57]
Randomly oriented squares 0.62554075(4),[51] 0.6254(2)[57] 0.9822723(1),[51] 0.9819(6)[57] 0.982278(14) [58] 0.9822723(1),[51] 0.9819(6)[57] 0.982278(14) [58]
Rectangles, ε = 1.1 0.624870(7) 0.980484(19) 1.078532(21) [58]
Rectangles, ε = 2 0.590635(5) 0.893147(13) 1.786294(26) [58]
Rectangles, ε = 3 0.5405983(34) 0.777830(7) 2.333491(22) [58]
Rectangles, ε = 4 0.4948145(38) 0.682830(8) 2.731318(30) [58]
Rectangles, ε = 5 0.4551398(31) 0.607226(6) 3.036130(28) [58]
Rectangles, ε = 10 0.3233507(25) 0.3906022(37) 3.906022(37) [58]
Rectangles, ε = 20 0.2048518(22) 0.2292268(27) 4.584535(54) [58]
Rectangles, ε = 50 0.09785513(36) 0.1029802(4) 5.149008(20) [58]
Rectangles, ε = 100 0.0523676(6) 0.0537886(6) 5.378856(60) [58]
Rectangles, ε = 200 0.02714526(34) 0.02752050(35) 5.504099(69) [58]
Rectangles, ε = 1000 0.00559424(6) 0.00560995(6) 5.609947(60) [58]
Sticks of length \ell 5.6372858(6),[51] 5.63726(2) [59]
Power-law disks, x=2.05 0.993(1) [60] 4.90(1) 0.0380(6)
Power-law disks, x=2.25 0.8591(5) [60] 1.959(5) 0.06930(12)
Power-law disks, x=2.5 0.7836(4) [60] 1.5307(17) 0.09745(11)
Power-law disks, x=4 0.69543(6) [60] 1.18853(19) 0.18916(3)
Power-law disks, x=5 0.68643(13) [60] 1.1597(3) 0.22149(8)
Power-law disks, x=6 0.68241(8) [60] 1.1470(1) 0.24340(5)
Power-law disks, x=7 0.6803(8) [60] 1.140(6) 0.25933(16)
Power-law disks, x=8 0.67917(9) [60] 1.1368(5) 0.27140(7)
Power-law disks, x=9 0.67856(12) [60] 1.1349(4) 0.28098(9)
Voids around disks of radius r 0.159(2) [61]

\eta_c = \pi r^2 N / L^2 equals critical total area for disks, where N is the number of objects and L is the system size.

\eta_c = \pi a b N / L^2 for ellipses of semi-major and semi-minor axes of a and b, respectively. Aspect ratio \epsilon = a / b with a > b.

\eta_c =  \ell m N / L^2 for rectangles of dimensions \ell and m. Aspect ratio \epsilon = \ell/m with \ell > m.

\eta_c = \pi x N / (4 L^2 (x-2)) for power-law distributed disks with \hbox{Prob(radius}\ge R) = R^{-x},  R \ge 1 .

\phi_c = 1 - e^{-\eta_c} equals critical area fraction.

n_c = \ell^2 N / L^2 equals number of objects of maximum length \ell = 2 a per unit area.

For ellipses, n_c =  (4 \epsilon / \pi)\eta_c

For void percolation, \phi_c = e^{-\eta_c} is the critical void fraction.

For more ellipse values, see [54]

For more rectangle values, see [58]

Thresholds on 2D random and quasi-lattices[edit]

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) [62] 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 \overline z Site Percolation Threshold Bond Percolation Threshold
Relative neighborhood graph 2.5576 0.796(2) [62] 0.771(2) [62]
Voronoi tessellation 3 0.71410(2),[63] 0.7151* [34] 0.68,[64] 0.666931(5),[63] 0.6670(1) [65]
Voronoi covering/medial 4 0.666931(2)[63][65] 0.53618(2) [63]
Penrose rhomb dual 4 0.6381(3)[30] 0.5233(2) [30]
Gabriel graph 4 0.6348(8),[66] 0.62[67] 0.5167(6),[66] 0.52[67]
Penrose rhomb 4 0.5837(3),[30] 0.58391(1)[68] 0.4770(2) [30]
Delaunay triangulation 6 1/2 [69] 0.333069(2) [63][65]
Octagonal lattice, "chemical" links (Ammann Beenker tiling) 4 0.585 [70] 0.48 [70]
Octagonal lattice, "ferromagnetic" links 5.17 0.543 [70] 0.40 [70]
Dodecagonal lattice, "chemical" links 3.63 0.628 [70] 0.54 [70]
Dodecagonal lattice, "ferromagnetic" links 4.27 0.617 [70] 0.495 [70]

*Theoretical estimate

Thresholds on slabs[edit]

Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
h= 2, SC, open b.c. 0.47424 [71]
h = 3, BCC, periodic b.c. 0.21113018(38) [72]
h = 4, BCC, periodic b.c. 0.20235168(59) [72]
h= 4, SC, open b.c. 0.3997 [71]
h = 5, SC, periodic b.c. 0.278102(5) [72]
h = 6, SC, periodic b.c. 0.272380(2) [72]
h = 7, SC, periodic b.c. 5,6 5,6 0.3459514(12) [72] 0.268459(1) [72]
h= 8, SC, open b.c. 0.3557 [71]
h = 8, SC, periodic b.c. 0.265615(5) [72]

More for SC open b.c. in Ref.[71]

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

Thresholds on 3D lattices[edit]

Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold Dimer Percolation Threshold
(10,3)-a oxide (or site-bond) [73] 2.4 0.748713(22)[73]
(10,3)-b oxide (or site-bond) [73] 2.4 0.745317(25)[73]
silicon dioxide (diamond site-bond) [73] 2 ⅔ 0.638683(35)[73]
(8,3)-a[74] 3 3 0.577962(33)[74] 0.555700(22)[74]
(10,3)-a[74] 3 3 0.571404(40)[74] 0.551060(37)[74]
(10,3)-b[74] 3 3 0.565442(40)[74] 0.546694(33)[74]
cubic oxide (cubic site-bond)[73] 3 0.524652(50)[73]
ice 4 4 0.433(11)[75] 0.388(10)[76]
diamond 4 4 0.4299870(4),[77] 0.426(+0.08,-0.02),[78] 0.4301(4),[79] 0.3895892(5),[77] 0.390(11),[76] 0.3893(2)[79]
kagome 6 0.3895(2) [80]
pentagonal stack 5⅓ 0.3394(4) [23] 0.2793(4) [23]
5⅓ 0.3480(4) [23] 0.2853(4) [23]
honeycomb stack 5 5 0.3701(2) [23] 0.3093(2) [23]
5 5 0.3840(4) [23] 0.3168(4) [23]
4 4 0.4560(6) [23] 0.4031(6) [23]
kagome stack 6 6 0.3346(4) [23] 0.2563(2) [23]
dice stack 6 6 0.2998(4) [23] 0.2378(4) [23]
bowtie stack 7 7 0.2822(6) [23] 0.2092(4) [23]
octagonal stack 8 8 0.2524(6) [23] 0.1752(2) [23]
simple cubic 6 6 0.3116077(2),[81] 0.311604(6),[82] 0.311605(5),[83] 0.311600(5),[84] 0.3116077(4),[85] 0.3116081(13),[86] 0.3116080(4),[87] 0.3116004(35),[88] 0.31160768(15)[77] 0.24881182(10),[81] 0.2488125(25),[89]

0.2488126(5) [90]

0.2555(1)[91]
Icosahedral Penrose 6 0.285[92] 0.225 [92]
Penrose w/2 diagonals 6.764 0.271[92] 0.207 [92]
Stacked triangular / simple hexagonal 8 8 0.26240(5),[93] 0.2625(2),[94] 0.2623(2)[23] 0.18602(2),[93] 0.1859(2) [23]
bcc 8 8 0.2459615(10),[87] 0.2460(3),[95] 0.2464(7),[96] 0.2458(2)[79] 0.1802875(10)[90]
simple cubic with 3NN 8 8 0.2455(1) [97]
fcc 12 12 0.1992365(10),[87] 0.19923517(20),[77] 0.1994(2)[79] 0.1201635(10)[90]
hcp 12 12 0.1992555(10)[98] 0.1201640(10)[98]
La2-x Srx Cu O4 12 12 0.19927(2) [99]
simple cubic with 2NN 12 12 0.1991(1) [97]
Penrose w/8 diagonals 12.764 0.188[92] 0.111 [92]
simple cubic with NN+3NN 14 14 0.1420(1) [97]
simple cubic with NN+2NN 18 18 0.1372(1),[97] 0.13735(5) [100]
simple cubic with short-length correlation 6+ 6+ 0.126(1)[101]
simple cubic with 2NN+3NN 20 20 0.1036(1) [97]
simple cubic with NN+2NN+3NN 26 26 0.0976(1),[97] 0.0976445(10) [100]

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

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?

System polymer Φc
percolating excluded volume of athermal polymer matrix (bond-fluctuation model on cubic lattice) 0.4304(3) [102]

Thresholds for 3D continuum models[edit]

All overlapping except for jammed spheres and polymer matrix.

System Φc ηc
Spheres of radius r 0.289573(2),[103] 0.2854 [104] 0.341889(3),[103] 0.3360[104]
Oblate ellipsoids with major radius r and aspect ratio 4/3 0.2831 [104] 0.3328[104]
Prolate ellipsoids with minor radius r and aspect ratio 3/2 0.2795 [104] 0.3278[104]
Oblate ellipsoids with major radius r and aspect ratio 2 0.2629 [104] 0.3050[104]
Prolate ellipsoids with minor radius r and aspect ratio 2 0.2618,[104] 0.25(2)[105] 0.3035,[104] 0.29(3) [105]
Oblate ellipsoids with major radius r and aspect ratio 3 0.2289 [104] 0.2599[104]
Prolate ellipsoids with minor radius r and aspect ratio 3 0.2244,[104] 0.20(2)[105] 0.2541,[104] 0.22(3)[105]
Oblate ellipsoids with major radius r and aspect ratio 4 0.2003 [104] 0.2235[104]
Prolate ellipsoids with minor radius r and aspect ratio 4 0.1901,[104] 0.16(2)[105] 0.2108,[104] 0.17(3)[105]
Oblate ellipsoids with major radius r and aspect ratio 5 0.1757 [104] 0.1932[104]
Prolate ellipsoids with minor radius r and aspect ratio 5 0.1627,[104] 0.13(2)[105] 0.1776,[104] 0.15(2)[105]
Oblate ellipsoids with major radius r and aspect ratio 10 0.1058 [104] 0.1118[104]
Prolate ellipsoids with minor radius r and aspect ratio 10 0.08703,[104] 0.07(2)[105] 0.09105,[104] 0.07(2)[105]
Oblate ellipsoids with major radius r and aspect ratio 100 0.01248[104] 0.01256[104]
Prolate ellipsoids with minor radius r and aspect ratio 100 0.006949[104] 0.006973[104]
Oblate ellipsoids with major radius r and aspect ratio 1000 0.001275 [104] 0.001276 [104]
Oblate ellipsoids with major radius r and aspect ratio 2000 0.000637[104] 0.000637 [104]
Aligned cylinders 0.2819(2)[106] 0.3312(1)[106]
Aligned cubes of side \ell = 2 a 0.2773(2) [57] 0.3247(3),[56] 0.3248(3)[57]
Randomly oriented icosahedra 0.3030(5) [107]
Randomly oriented dodecahedra 0.2949(5) [107]
Randomly oriented octahedra 0.2514(6) [107]
Randomly oriented cubes of side \ell = 2 a 0.2168(2) [57] 0.2444(3),[57] 0.2443(5)[107]
Randomly oriented tetrahedra 0.1701(7) [107]
Randomly oriented disks of radius r (in 3D) 0.9614(5)[108]
Randomly oriented square plates of side \sqrt{\pi} r 0.8647(6)[108]
Randomly oriented triangular plates of side \sqrt{2 \pi} /3^{1/4} r 0.7295(6)[108]
Voids around disks of radius r 22.86(2)[109]
Voids around oblate ellipsoids of major radius r and aspect ratio 10 15.42(1)[109]
Voids around oblate ellipsoids of major radius r and aspect ratio 2 6.478(8)[109]
Voids around spheres of radius r 0.030(2),[61] 0.0301(3),[110] 0.0294,[111] 0.0300(3) [112] 0.0317(4) [113] 3.506(8),[112] 3.515(6) [109]
Jammed spheres (average z = 6) 0.183(3)[114] 0.59(1) [114]

\eta_c = (4/3) \pi r^3 N / L^3 is the total volume, where N is the number of objects and L is the system size.

\phi_c = 1 - e^{-\eta_c} is the critical volume fraction.

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

For void ("Swiss-Cheese" model), \phi_c = e^{-\eta_c} is the critical void fraction.

For more results on void percolation around ellipsoids and elliptical plates, see.[109]

For more ellipsoid percolation values see [104]

Thresholds on hypercubic lattices[edit]

d z Site Thresholds Bond Thresholds
4 8 0.1968861(14),[115] 0.196889(3),[116] 0.196901(5) [117] 0.1601314(13),[115] 0.160130(3),[116] 0.1601310(10) [89]
5 10 0.1407966(15) [115] 0.118172(1),[115] 0.1181718(3) [89]
6 12 0.109017(2) [115] 0.0942019(6) [115]
7 14 0.0889511(9),[115] 0.088939(20) [118] 0.0786752(3) [115]
8 16 0.0752101(5) [115] 0.06770839(7) [115]
9 18 0.0652095(3) [115] 0.05949601(5) [115]
10 20 0.0575930(1) [115] 0.05309258(4) [115]
11 22 0.05158971(8) [115] 0.04794969(1) [115]
12 24 0.04673099(6) [115] 0.04372386(1) [115]
13 26 0.04271508(8) [115] 0.04018762(1) [115]
d z Site Thresholds Bond Thresholds τ
4 8 0.196889(3) [116] 0.160130(3) [116] 2.313(3) [116]
5 10 0.14081(1) [116] 0.118174(4) [116] 2.412(4) [116]

Simulation parameters and results for pc and the Fisher exponent τ.

d z Site Thresholds Bond Thresholds zspread dmin
4 8 0.196889 [116] 0.160130 [116] 0.622(2) [116] 1.607(5) [116]
5 10 0.14081 [116] 0.118174 [116] 0.552(2) [116] 1.812(6) [116]

Simulation parameters and results for the spreading exponent zspread and shortest path exponent.

Thresholds in higher-dimensional lattices[edit]

d lattice z Site Thresholds Bond Thresholds
4 diamond 5 0.2978(2)[79]
4 kagome 8 0.2715(3) [80]
4 fcc 24 0.0842(3)[79] 0.049(1)[79]
5 diamond 6 0.2252(3)[79]
5 kagome 10 0.2084(4) [80]
5 bcc 32 0.0446(4)[79] 0.033(1)[79]
5 fcc 40 0.0431(3)[79] 0.026(2)[79]
6 diamond 7 0.1799(5)[79]
6 kagome 12 0.1677(7) [80]
6 fcc 60 0.0252(5)[79]
6 bcc 64 0.0199(5)[79]

Thresholds on hyperbolic, hierarchical, and tree lattices[edit]

Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk [119]
Depiction of the non-planar Hanoi network HN-NP [120]


Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
Lower Upper
{4,5} hyperbolic 5 5 0.27[121] 0.52[121]
{7,3} hyperbolic 3 3 0.72[121] 0.53[121]
{3,7} hyperbolic 7 7 0.20[121] 0.37[121]
{∞,3} Cayley tree 3 3 1/2 1/2[121] 1[121]
Enhanced binary tree (EBT) 0.304(1)[121] 0.48,[121] 0.564(1)[122]
Enhanced binary tree dual 0.436(1)[122] 0.696(1)[122]
Non-Planar Hanoi Network (HN-NP) 0.319445[120] 0.381996[120]
Cayley tree with grandparents 8 0.158656326[123]

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

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

Cayley tree with a distribution of z with mean  \overline z , mean-square  \overline{z^2}:  pc=  \overline z / (\overline{z^2} - \overline z) [124] (site or bond threshold)

Thresholds for directed percolation[edit]

(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),[125] 0.839933(5),[126] 0.8228569(2),[125] 0.82285680(6)[125]
(1+1)-d kagome 2 0.7369317(2),[125] 0.73693182(4)[127] 0.6589689(2),[125] 0.65896910(8)[125]
(1+1)-d square, diagonal direction 2 0.705489(4),[128] 0.70548522(4),[129] 0.70548515(20),[127]

0.7054852(3),[125]

0.644701(2),[130] 0.644701(1),[131] 0.64470015(5),[132] 0.644700185(5),[129] 0.6447001(2),[125]
(1+1)-d triangular 3 0.5956468(5),[132] 0.5956470(3) [125] 0.478025(1),[132] 0.4780250(4) [125]
(2+1)-d simple cubic, diagonal planes 3 0.43531(1) [133] 0.382223(7) [133]
(2+1)-d square nn (= bcc) 4 0.3445736(3),[134] 0.344575(15) [135] 0.2873383(1),[136] 0.287338(3)[133]
(3+1)-d hypercubic, diagonal planes 4 0.3025(10) [137]
(3+1)-d cubic, nn 6 0.2081040(4) [134] 0.1774970(5) [89]
(3+1)-d body-centered hypercubic 8 0.160950(30) [135]
(4+1)-d hypercubic, nn 8 0.1461593(2),[134] 0.1461582(3) [138] 0.1288557(5) [89]
(4+1)-d body-centered hypercubic 16 0.075582(17) [135]

0.0755850(3) [138]

(5+1)-d hypercubic, nn 10 0.1123373(2) [134] 0.1016796(5) [89]
(5+1)-d body-centered hypercubic 32 0.035967(23) [135]
(6+1)-d hypercubic, nn 12 0.0913087(2) [134] 0.0841997(14) [89]
(7+1)-d hypercubic,nn 14 0.07699336(7) [134] 0.07195(5) [89]

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[edit]

Inhomogeneous triangular lattice bond percolation[13]


1 - p_1  - p_2 - p_3 + p_1 p_2 p_3 = 0

Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation[13]


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[4] [139]


1 - 3(s_1s_2)^2 + (s_1s_2)^3 = 0,
or 
s_1 s_2 = 1 - 2 \sin(\pi/18)

Inhomogeneous martini lattice, bond percolation [39] 
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

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

Inhomogeneous martini-A (3–7) lattice, bond percolation. Left side (top of "A" to bottom): r_2,\  p_1. Right side: r_1, \  p_2. Cross bond: \ 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

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

Inhomogeneous checkerboard lattice, bond percolation [28][50]


 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 [27][50]


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 are the four bonds around the square and u is the diagonal bond connecting the vertex between bonds p_4, p_1 and p_2, p_3.

Percolation thresholds of graphs[edit]

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>.[140] The critical threshold was calculated exactly also for interdependent ER networks.[141]

See also[edit]

References[edit]

  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. 
  2. ^ a b c d e Grünbaum, Branko; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. 
  3. ^ a b c d e f g Parviainen, Robert (2005). Connectivity Properties of Archimedean and Laves Lattices 34. Uppsala Dissertations in Mathematics. p. 37. ISBN 91-506-1751-6. 
  4. ^ a b c d e f g h i Suding, P. N.; R. M. Ziff (1999). "Site percolation thresholds for Archimedean lattices". Physical Review E 60 (1): 275–283. Bibcode:1999PhRvE..60..275S. doi:10.1103/PhysRevE.60.275. 
  5. ^ a b c d e f g h i j k l m n o p q r s t u Jacobsen, J. L. (2014). "High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials". J. Phys. A: Math. Theor. 47 (13): 135001. arXiv:1401.7847. Bibcode:2014JPhA...47m5001G. doi:10.1088/1751-8113/47/13/135001. 
  6. ^ a b Jacobsen, Jesper L.; Christian R. Scullard (2013). "Critical manifolds, graph polynomials, and exact solvability". StatPhys 25, Seoul, Korea July 21–26 http://www.statphys25.org/data/Statphys25%20Abstract%20Book.pdf. 
  7. ^ a b Scullard, C. R.; J. L. Jacobsen (2012). Transfer matrix computation of generalised critical polynomials in percolation. arXiv:1209.1451. Bibcode:2012arXiv1209.1451S. 
  8. ^ a b c d e f g Parviainen, Robert (2007). "Estimation of bond percolation thresholds on the Archimedean lattices". J. Phys. A 40 (31): 9253–9258. arXiv:0704.2098. Bibcode:2007JPhA...40.9253P. doi:10.1088/1751-8113/40/31/005. 
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