The percolation threshold is a mathematical concept in percolation theory that describes 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.[1]

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 = \frac{1}{g_1'(1)} = \frac{\langle k \rangle}{\langle k^2 \rangle - \langle k \rangle}.

Where g_1(z) is the generating function corresponding to the excess degree distribution, {\langle k \rangle} is the average degree of the network and {\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{\langle k^2 \rangle = \langle k \rangle^2 + \langle k \rangle}, the threshold is at p_c = {\langle k \rangle}^{-1}.

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

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

Latticez\overline zSite percolation thresholdBond percolation threshold
3-12 or super-kagome, (3, 122 )330.807900764... = (1 − 2 sin (π/18))1⁄2[6]0.74042195(80),[7] 0.74042077(2),[8] 0.740420800(2),[9] 0.7404207988509(8),[10][11] 0.740420798850811610(2),[12]
cross, truncated trihexagonal (4, 6, 12)330.746,[13] 0.750,[14] 0.747806(4),[6] 0.7478008(2)[10]0.6937314(1),[10] 0.69373383(72),[7] 0.693733124922(2)[12]
square octagon, bathroom tile, 4-8, truncated square (4, 82)33|-0.729,[13] 0.729724(3),[6] 0.7297232(5)[10]0.6768,[15] 0.67680232(63),[7] 0.6768031269(6),[10] 0.6768031243900113(3),[12]
honeycomb (63)330.6962(6),[16] 0.697040230(5),[10] 0.6970402(1),[17] 0.6970413(10),[18] 0.697043(3),[6]0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=0[19]
kagome (3, 6, 3, 6)440.652703645... = 1 − 2 sin(π/18)[19]0.5244053(3),[20] 0.52440516(10),[18] 0.52440499(2),[17] 0.524404978(5),[8] 0.52440572...,[21] 0.52440500(1),[9] 0.524404999173(3),[10][11] 0.524404999167439(4)[22] 0.52440499916744820(1)[12]
ruby,[23] rhombitrihexagonal (3, 4, 6, 4)440.620,[13] 0.621819(3),[6] 0.62181207(7)[10]0.52483258(53),[7] 0.5248311(1),[10] 0.524831461573(1)[12]
square (44)440.59274(10),[24] 0.59274605079210(2),[22] 0.59274601(2),[10] 0.59274605095(15),[25] 0.59274621(13),[26] 0.592746050786(3),[27] 0.5927460507896(1),[28] 0.59274605079016(1),[29] 0.59274621(33),[30] 0.59274598(4),[31][32] 0.59274605(3),[17] 0.593(1),[33] 0.591(1),[34] 0.569(13),[35] 0.59274(5)[36]1⁄2
snub hexagonal, maple leaf[37] (34,6)550.579[14] 0.579498(3)[6]0.43430621(50),[7] 0.43432764(3),[10] 0.4343283172240(6),[12]
snub square, puzzle, Shastry-Sutherland[38] (32, 4, 3, 4 )550.550,[13][39] 0.550806(3)[6]0.41413743(46),[7] 0.4141378476(7),[10] 0.4141378565917(1),[12]
frieze, trellis, elongated triangular(33, 42)550.549,[13] 0.550213(3),[6] 0.5502(8)[40]0.4196(6),[40] 0.41964191(43),[7] 0.41964044(1),[10] 0.41964035886369(2)[12]
triangular (36)661⁄20.347296355... = 2 sin (π/18), 1 + p3 − 3p = 0[19]

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),[41] etc. Equivalent to square-2N+3N+4N,[42] sq(1,2,3).[43] tri = triangular, hc = honeycomb.

LatticezSite percolation thresholdBond percolation threshold
sq-1, sq-2, sq-3, sq-540.5927...[41][42] (square site)
sq-1,2, sq-2,3, sq-3,5... 3x3 square80.407...[41][42][44] (square matching)0.25036834(6),[17] 0.2503685,[45] 0.25036840(4)[46]
sq-1,380.337[41][42]0.2214995[45]
sq-2,5: 2NN+5NN80.337[42]
hc-1,2,3: honeycomb-NN+2NN+3NN120.300,[43] 0.300,[14] 0.302960... = 1-pc(site, hc)[47]
tri-1,2: triangular-NN+2NN120.295,[43] 0.289,[14] 0.290258(19)[48]
tri-2,3: triangular-2NN+3NN120.232020(36),[49] 0.232020(20)[48]
sq-4: square-4NN80.270...[42]
sq-1,5: square-NN+5NN (r ≤ 2)80.277[42]
sq-1,2,3: square-NN+2NN+3NN120.292,[50] 0.290(5)[51] 0.289,[14] 0.288,[41][42] 0.2891226(14)[52]0.1522203[45]
sq-2,3,5: square-2NN+3NN+5NN120.288[42]
sq-1,4: square-NN+4NN120.236[42]
sq-2,4: square-2NN+4NN120.225[42]
tri-4: triangular-4NN120.192450(36),[49] 0.1924428(50)[48]
hc-2,4: honeycomb-2NN+4NN120.2374[53]
tri-1,3: triangular-NN+3NN120.264539(21)[48]
tri-1,2,3: triangular-NN+2NN+3NN180.225,[50] 0.215,[14] 0.215459(36)[49] 0.2154657(17)[48]
sq-3,4: 3NN+4NN120.221[42]
sq-1,2,5: NN+2NN+5NN120.240[42]0.13805374[45]
sq-1,3,5: NN+3NN+5NN120.233[42]
sq-4,5: 4NN+5NN120.199[42]
sq-1,2,4: NN+2NN+4NN160.219[42]
sq-1,3,4: NN+3NN+4NN160.208[42]
sq-2,3,4: 2NN+3NN+4NN160.202[42]
sq-1,4,5: NN+4NN+5NN160.187[42]
sq-2,4,5: 2NN+4NN+5NN160.182[42]
sq-3,4,5: 3NN+4NN+5NN160.179[42]
sq-1,2,3,5 asterisk pattern160.208[42]0.1032177[45]
tri-4,5: 4NN+5NN180.140250(36),[49]
sq-1,2,3,4: NN+2NN+3NN+4NN (r \le \sqrt{5})200.19671(9),[54] 0.196,[42] 0.196724(10),[55] 0.1967293(7)[52]0.0841509[45]
sq-1,2,4,5: NN+2NN+4NN+5NN200.177[42]
sq-1,3,4,5: NN+3NN+4NN+5NN200.172[42]
sq-2,3,4,5: 2NN+3NN+4NN+5NN200.167[42]
sq-1,2,3,5,6 asterisk pattern200.0783110[45]
sq-1,2,3,4,5: NN+2NN+3NN+4NN+5NN (r \le \sqrt{8}, also within a 5 × 5 square)240.164,[14] 0.164,[42] 0.1647124(6)[52]
sq-1,2,3,4,6: NN+2NN+3NN+4NN+6NN (diamond r \le 3)240.16134,[56]
tri-1,4,5: NN+4NN+5NN240.131660(36)[49]
sq-1,...,6: NN+...+6NN (r≤3)280.142,[14] 0.1432551(9)[52]0.0558493[45]
tri-2,3,4,5: 2NN+3NN+4NN+5NN300.117460(36)[49] 0.135823(27)[48]
tri-1,2,3,4,5: NN+2NN+3NN+4NN+5NN
360.115,[14] 0.115740(36),[49] 0.1157399(58)[48]
sq-1,...,7: NN+...+7NN (r \le \sqrt{10})360.113,[14] 0.1153481(9)[52]0.04169608[45]
sq lat, diamond boundary: dist. ≤ 4400.105(5)[51]
sq-1,...,8: NN+..+8NN (r \le \sqrt{13})440.095,[39] 0.095765(5),[55] 0.09580(2),[54] 0.0957661(9)[52]
sq-1,...,9: NN+..+9NN (r≤4)480.086[14]0.02974268[45]
sq-1,...,11: NN+...+11NN (r \le \sqrt{18})600.02301190(3)[45]
sq-1,...,23 (r ≤ 7)1480.008342595[46]
sq-1,...,32: NN+...+32NN (r \le \sqrt{72})2240.0053050415(33)[45]
sq-1,...,86: NN+...+86NN (r≤15)7080.001557644(4)[57]
sq-1,...,141: NN+...+141NN (r \le \sqrt{389})12240.000880188(90)[45]
sq-1,...,185: NN+...+185NN (r≤23)16520.000645458(4)[57]
sq-1,...,317: NN+...+317NN (r≤31)30000.000349601(3)[57]
sq-1,...,413: NN+...+413NN (r \le \sqrt{1280})40160.0002594722(11)[45]
sq lat, diamond boundary: dist. ≤ 6840.049(5)[51]
sq lat, diamond boundary: dist. ≤ 81440.028(5)[51]
sq lat, diamond boundary: dist. ≤ 102200.019(5)[51]
2x2 touching lattice squares* (same as sq-1,2,3,4)20φc = 0.58365(2),[55] pc = 0.196724(10),[55] 0.19671(9),[54]
3x3 touching lattice squares* (same as sq-1,...,8))44φc = 0.59586(2),[55] pc = 0.095765(5),[55] 0.09580(2)[54]
4x4 touching lattice squares*76φc = 0.60648(1),[55] pc = 0.0566227(15),[55] 0.05665(3),[54]
5x5 touching lattice squares*116φc = 0.61467(2),[55] pc = 0.037428(2),[55] 0.03745(2),[54]
6x6 touching lattice squares*220pc = 0.02663(1),[54]
10x10 touching lattice squares*436φc = 0.63609(2),[55] pc = 0.0100576(5)[55]
within 11 × 11 square (r=5)1200.01048079(6)[57]
within 15 × 15 square (r=7)2240.005287692(22)[57]
20x20 touching lattice squares*1676φc = 0.65006(2),[55] pc = 0.0026215(3)[55]
within 31 × 31 square (r=15)9600.001131082(5)[57]
100x100 touching lattice squares*40396φc = 0.66318(2),[55] pc = 0.000108815(12)[55]
1000x1000 touching lattice squares*4003996φc = 0.66639(1),[55] pc = 1.09778(6)E-06 [55]

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.[41]

  • For overlapping or touching squares, p_c(site) given here is the net fraction of sites occupied \phi_c similar to the \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-\phi_c)^{1/4} = 0.196724(10)\ldots with \phi_c= 0.58365(2).[55] The 3×3 square corresponds to sq-1,2,3,4,5,6,7,8 with z=44 and p_c=1-(1-\phi_c)^{1/9} = 0.095765(5)\ldots. The value of z for a k × 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-\alpha,x+\alpha),(y-\alpha,y+\alpha), and considers percolation when sites are within Euclidean distance d of each other.

Lattice\overline z\alphadSite percolation thresholdBond percolation threshold
square0.21.10.8025(2)[58]
0.21.20.6667(5)[58]
0.11.10.6619(1)[58]

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+10k-2 for 1 \times k sticks.

SystemkzSite coverage φcSite percolation threshold pc
1 × 2 dimer, square lattice2220.54691[54] 0.5483(2)[59]0.17956(3)[54] 0.18019(9)[59]
1 × 2 aligned dimer, square lattice214(?)0.5715(18)[59]0.3454(13)[59]
1 × 3 trimer, square lattice3370.49898[54] 0.50004(64)[59]0.10880(2)[54] 0.1093(2)[59]
1 × 4 stick, square lattice4540.45761[54]0.07362(2)[54]
1 × 5 stick, square lattice5730.42241[54]0.05341(1)[54]
1 × 6 stick, square lattice6940.39219[54]0.04063(2)[54]

The coverage is calculated from p_c by \phi_c = 1-(1-p_c)^{2 k} for 1 \times k sticks, because there are 2k sites where a stick will cause an overlap with a given site.

For aligned 1 \times k sticks: \phi_c = 1-(1-p_c)^{k}

Approximate formulas for thresholds of Archimedean lattices

LatticezSite percolation thresholdBond percolation threshold
(3, 122 )3
(4, 6, 12)3
(4, 82)30.676835..., 4p3 + 3p4 − 6 p5 − 2 p6 = 1[60]
honeycomb (63)3
kagome (3, 6, 3, 6)40.524430..., 3p2 + 6p3 − 12 p4+ 6 p5 − p6 = 1[61]
(3, 4, 6, 4)4
square (44)41⁄2 (exact)
(34,6 )50.434371..., 12p3 + 36p4 − 21p5 − 327 p6 + 69p7 + 2532p8 − 6533 p9 + 8256 p10 − 6255p11 + 2951p12 − 837 p13 + 126 p14 − 7p15 = 1 [citation needed]
snub square, puzzle (32, 4, 3, 4 )5
(33, 42)5
triangular (36)61⁄2 (exact)

AB percolation and colored percolation in 2D

In AB percolation, a p_\mathrm{site} is the proportion of A sites among B sites, and bonds are drawn between sites of opposite species.[62] It is also called antipercolation.

In colored percolation, occupied sites are assigned one of n colors with equal probability, and connection is made along bonds between neighbors of different colors.[63]

Latticez\overline zSite percolation threshold
triangular AB660.2145,[62] 0.21524(34),[64] 0.21564(3)[65]
AB on square-covering lattice661-\sqrt{1-p_c(site,sq)} = 0.361835[66]
square three-color440.80745(5)[63]
square four-color440.73415(4)[63]
square five-color440.69864(7)[63]
square six-color440.67751(5)[63]
triangular two-color660.72890(4)[63]
triangular three-color660.63005(4)[63]
triangular four-color660.59092(3)[63]
triangular five-color660.56991(5)[63]
triangular six-color660.55679(5)[63]

Site-bond percolation in 2D

Site bond percolation. Here p_s is the site occupation probability and 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}) = 0, and some specific critical pairs (p_{s},p_{b}) are listed below.

Square lattice:

Latticez\overline zSite percolation thresholdBond percolation threshold
square440.615185(15)[67]0.95
0.667280(15)[67]0.85
0.732100(15)[67]0.75
0.750.726195(15)[67]
0.815560(15)[67]0.65
0.850.615810(30)[67]
0.950.533620(15)[67]

Honeycomb (hexagonal) lattice:

Latticez\overline zSite percolation thresholdBond percolation threshold
honeycomb330.7275(5)[68]0.95
0. 0.7610(5)[68]0.90
0.7986(5)[68]0.85
0.800.8481(5)[68]
0.8401(5)[68]0.80
0.850.7890(5)[68]
0.900.7377(5)[68]
0.950.6926(5)[68]

Kagome lattice:

Latticez\overline zSite percolation thresholdBond percolation threshold
kagome440.6711(4),[68] 0.67097(3)[69]0.95
0.6914(5),[68] 0.69210(2)[69]0.90
0.7162(5),[68] 0.71626(3)[69]0.85
0.7428(5),[68] 0.74339(3)[69]0.80
0.750.7894(9)[68]
0.7757(8),[68] 0.77556(3)[69]0.75
0.800.7152(7)[68]
0.81206(3)[69]0.70
0.850.6556(6)[68]
0.85519(3)[69]0.65
0.900.6046(5)[68]
0.90546(3)[69]0.60
0.950.5615(4)[68]
0.96604(4)[69]0.55
0.9854(3)[69]0.53

* For values on different lattices, see "An investigation of site-bond percolation on many lattices".[68]

Approximate formula for site-bond percolation on a honeycomb lattice

Latticez\overline zThresholdNotes
(63) honeycomb33p_b p_s [1 - (\sqrt{p_{bc}}/(3-p_{bc}))(\sqrt{p_b} - \sqrt{p_{bc}})] = p_{bc}, When equal: ps = pb = 0.82199approximate formula, ps = site prob., pb = bond prob., pbc = 1 − 2 sin (π/18),[18] exact at ps=1, pb=pbc.

Archimedean duals (Laves lattices)

Laves lattices are the duals to the Archimedean lattices. Drawings from.[70] See also Uniform tilings.

Latticez\overline zSite percolation thresholdBond percolation threshold
Cairo pentagonal D(32,4,3,4)=(2⁄3)(53)+(1⁄3)(54)3,43 1⁄30.6501834(2),[10] 0.650184(5)[70]0.585863... = 1 − pcbond(32,4,3,4)
Pentagonal D(33,42)=(1⁄3)(54)+(2⁄3)(53)3,43 1⁄30.6470471(2),[10] 0.647084(5),[70] 0.6471(6)[40]0.580358... = 1 − pcbond(33,42), 0.5800(6)[40]
D(34,6)=(1⁄5)(46)+(4⁄5)(43)3,63 3⁄50.639447[70]0.565694... = 1 − pcbond(34,6 )
dice, rhombille tiling D(3,6,3,6) = (1⁄3)(46) + (2⁄3)(43)3,640.5851(4),[71] 0.585040(5)[70]0.475595... = 1 − pcbond(3,6,3,6 )
ruby dual D(3,4,6,4) = (1⁄6)(46) + (2⁄6)(43) + (3⁄6)(44)3,4,640.582410(5)[70]0.475167... = 1 − pcbond(3,4,6,4 )
union jack, tetrakis square tiling D(4,82) = (1⁄2)(34) + (1⁄2)(38)4,861⁄20.323197... = 1 − pcbond(4,82 )
bisected hexagon,[72] cross dual D(4,6,12)= (1⁄6)(312)+(2⁄6)(36)+(1⁄2)(34)4,6,1261⁄20.306266... = 1 − pcbond(4,6,12)
asanoha (hemp leaf)[73] D(3, 122)=(2⁄3)(33)+(1⁄3)(312)3,1261⁄20.259579... = 1 − pcbond(3, 122)

2-uniform lattices

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

[3]

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

[3]

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

[3]

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

[3]

Latticez\overline zSite percolation thresholdBond percolation threshold
41(1⁄2)(3,4,3,12) + (1⁄2)(3, 122)4,33.50.7680(2)[74]0.67493252(36)[citation needed]
42(1⁄3)(3,4,6,4) + (2⁄3)(4,6,12)4,331⁄30.7157(2)[74]0.64536587(40)[citation needed]
36(1⁄7)(36) + (6⁄7)(32,4,12)6,44 2⁄70.6808(2)[74]0.55778329(40)[citation needed]
15(2⁄3)(32,62) + (1⁄3)(3,6,3,6)4,440.6499(2)[74]0.53632487(40)[citation needed]
34(1⁄7)(36) + (6⁄7)(32,62)6,44 2⁄70.6329(2)[74]0.51707873(70)[citation needed]
16(4⁄5)(3,42,6) + (1⁄5)(3,6,3,6)4,440.6286(2)[74]0.51891529(35)[citation needed]
17(4⁄5)(3,42,6) + (1⁄5)(3,6,3,6)*4,440.6279(2)[74]0.51769462(35)[citation needed]
35(2⁄3)(3,42,6) + (1⁄3)(3,4,6,4)4,440.6221(2)[74]0.51973831(40)[citation needed]
11(1⁄2)(34,6) + (1⁄2)(32,62)5,44.50.6171(2)[74]0.48921280(37)[citation needed]
37(1⁄2)(33,42) + (1⁄2)(3,4,6,4)5,44.50.5885(2)[74]0.47229486(38)[citation needed]
30(1⁄2)(32,4,3,4) + (1⁄2)(3,4,6,4)5,44.50.5883(2)[74]0.46573078(72)[citation needed]
23(1⁄2)(33,42) + (1⁄2)(44)5,44.50.5720(2)[74]0.45844622(40)[citation needed]
22(2⁄3)(33,42) + (1⁄3)(44)5,44 2⁄30.5648(2)[74]0.44528611(40)[citation needed]
12(1⁄4)(36) + (3⁄4)(34,6)6,55 1⁄40.5607(2)[74]0.41109890(37)[citation needed]
33(1⁄2)(33,42) + (1⁄2)(32,4,3,4)5,550.5505(2)[74]0.41628021(35)[citation needed]
32(1⁄3)(33,42) + (2⁄3)(32,4,3,4)5,550.5504(2)[74]0.41549285(36)[citation needed]
31(1⁄7)(36) + (6⁄7)(32,4,3,4)6,55 1⁄70.5440(2)[74]0.40379585(40)[citation needed]
13(1⁄2)(36) + (1⁄2)(34,6)6,55.50.5407(2)[74]0.38914898(35)[citation needed]
21(1⁄3)(36) + (2⁄3)(33,42)6,55 1⁄30.5342(2)[74]0.39491996(40)[citation needed]
20(1⁄2)(36) + (1⁄2)(33,42)6,55.50.5258(2)[74]0.38285085(38)[citation needed]

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 condition[75] 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):

Latticez\overline zSite percolation thresholdBond percolation threshold
martini (3⁄4)(3,92)+(1⁄4)(93)330.764826..., 1 + p4 − 3p3 = 0[76]0.707107... = 1/√2[77]
bow-tie (c)3,43 1⁄70.672929..., 1 − 2p3 − 2p4 − 2p5 − 7p6 + 18p7 + 11p8 − 35p9 + 21p10 − 4p11 = 0[78]
bow-tie (d)3,43 1⁄30.625457..., 1 − 2p2 − 3p3 + 4p4 − p5 = 0[78]
martini-A (2⁄3)(3,72)+(1⁄3)(3,73)3,43 1⁄31/√2[78]0.625457..., 1 − 2p2 − 3p3 + 4p4 − p5 = 0[78]
bow-tie dual (e)3,43 2⁄30.595482..., 1-pcbond (bow-tie (a))[78]
bow-tie (b)3,4,63 2⁄30.533213..., 1 − p − 2p3 -4p4-4p5+156+ 13p7-36p8+19p9+ p10 + p11=0[78]
martini covering/medial (1⁄2)(33,9) + (1⁄2)(3,9,3,9)440.707107... = 1/√2[77]0.57086651(33)[citation needed]
martini-B (1⁄2)(3,5,3,52) + (1⁄2)(3,52)3, 540.618034... = 2/(1 + √5), 1- p2 − p = 0[76][78]1⁄2[77][78]
bow-tie dual (f)3,4,84 2⁄50.466787..., 1 − pcbond (bow-tie (b))[78]
bow-tie (a) (1⁄2)(32,4,32,4) + (1⁄2)(3,4,3)4,650.5472(2),[40] 0.5479148(7)[79]0.404518..., 1 − p − 6p2 + 6p3 − p5 = 0[78][80]
bow-tie dual (h)3,6,850.374543..., 1 − pcbond(bow-tie (d))[78]
bow-tie dual (g)3,6,105 1⁄20.547... = pcsite(bow-tie(a))0.327071..., 1 − pcbond(bow-tie (c))[78]
martini dual (1⁄2)(33) + (1⁄2)(39)3,961⁄20.292893... = 1 − 1/√2[77]

Thresholds on 2D covering, medial, and matching lattices

Latticez\overline zSite percolation thresholdBond percolation threshold
(4, 6, 12) covering/medial44pcbond(4, 6, 12) = 0.693731...0.5593140(2),[10] 0.559315(1)[citation needed]
(4, 82) covering/medial, square kagome44pcbond(4,82) = 0.676803...0.544798017(4),[10] 0.54479793(34)[citation needed]
(34, 6) medial440.5247495(5)[10]
(3,4,6,4) medial440.51276[10]
(32, 4, 3, 4) medial440.512682929(8)[10]
(33, 42) medial440.5125245984(9)[10]
square covering (non-planar)661⁄20.3371(1)[60]
square matching lattice (non-planar)881 − pcsite(square) = 0.407253...0.25036834(6)[17]

Thresholds on 2D chimera non-planar lattices

Latticez\overline zSite percolation thresholdBond percolation threshold
K(2,2)440.51253(14)[81]0.44778(15)[81]
K(3,3)660.43760(15)[81]0.35502(15)[81]
K(4,4)880.38675(7)[81]0.29427(12)[81]
K(5,5)10100.35115(13)[81]0.25159(13)[81]
K(6,6)12120.32232(13)[81]0.21942(11)[81]
K(7,7)14140.30052(14)[81]0.19475(9)[81]
K(8,8)16160.28103(11)[81]0.17496(10)[81]

Thresholds on subnet lattices

The 2 × 2, 3 × 3, and 4 × 4 subnet kagome lattices. The 2 × 2 subnet is also known as the "triangular kagome" lattice.[82]

Latticez\overline zSite percolation thresholdBond percolation threshold
checkerboard – 2 × 2 subnet4,30.596303(1)[83]
checkerboard – 4 × 4 subnet4,30.633685(9)[83]
checkerboard – 8 × 8 subnet4,30.642318(5)[83]
checkerboard – 16 × 16 subnet4,30.64237(1)[83]
checkerboard – 32 × 32 subnet4,30.64219(2)[83]
checkerboard – \infty subnet4,30.642216(10)[83]
kagome – 2 × 2 subnet = (3, 122) covering/medial4pcbond (3, 122) = 0.74042077...0.600861966960(2),[10] 0.6008624(10),[18] 0.60086193(3)[8]
kagome – 3 × 3 subnet40.6193296(10),[18] 0.61933176(5),[8] 0.61933044(32)[citation needed]
kagome – 4 × 4 subnet40.625365(3),[18] 0.62536424(7)[8]
kagome – \infty subnet40.628961(2)[18]
kagome – (1 × 1):(2 × 2) subnet = martini covering/medial4pcbond(martini) = 1/√2 = 0.707107...0.57086648(36)[citation needed]
kagome – (1 × 1):(3 × 3) subnet4,30.728355596425196...[8]0.58609776(37)[citation needed]
kagome – (1 × 1):(4 × 4) subnet0.738348473943256...[8]
kagome – (1 × 1):(5 × 5) subnet0.743548682503071...[8]
kagome – (1 × 1):(6 × 6) subnet0.746418147634282...[8]
kagome – (2 × 2):(3 × 3) subnet0.61091770(30)[citation needed]
triangular – 2 × 2 subnet6,40.471628788[83]
triangular – 3 × 3 subnet6,40.509077793[83]
triangular – 4 × 4 subnet6,40.524364822[83]
triangular – 5 × 5 subnet6,40.5315976(10)[83]
triangular – \infty subnet6,40.53993(1)[83]

Thresholds of random sequentially adsorbed objects

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

systemzSite threshold
dimers on a honeycomb lattice30.69,[84] 0.6653 [85]
dimers on a triangular lattice60.4872(8),[84] 0.4873,[85]
aligned linear dimers on a triangular lattice60.5157(2)[86]
aligned linear 4-mers on a triangular lattice60.5220(2)[86]
aligned linear 8-mers on a triangular lattice60.5281(5)[86]
aligned linear 12-mers on a triangular lattice60.5298(8)[86]
linear 16-mers on a triangular lattice6aligned 0.5328(7)[86]
linear 32-mers on a triangular lattice6aligned 0.5407(6)[86]
linear 64-mers on a triangular lattice6aligned 0.5455(4)[86]
aligned linear 80-mers on a triangular lattice60.5500(6)[86]
aligned linear k \longrightarrow \infty on a triangular lattice60.582(9)[86]
dimers and 5% impurities, triangular lattice60.4832(7)[87]
parallel dimers on a square lattice40.5863[88]
dimers on a square lattice40.5617,[88] 0.5618(1),[89] 0.562,[90] 0.5713[85]
linear 3-mers on a square lattice40.528[90]
3-site 120° angle, 5% impurities, triangular lattice60.4574(9)[87]
3-site triangles, 5% impurities, triangular lattice60.5222(9)[87]
linear trimers and 5% impurities, triangular lattice60.4603(8)[87]
linear 4-mers on a square lattice40.504[90]
linear 5-mers on a square lattice40.490[90]
linear 6-mers on a square lattice40.479[90]
linear 8-mers on a square lattice40.474,[90] 0.4697(1)[89]
linear 10-mers on a square lattice40.469[90]
linear 16-mers on a square lattice40.4639(1)[89]
linear 32-mers on a square lattice40.4747(2)[89]

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.[91]

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.

systemzBond threshold
Parallel covering, square lattice60.381966...[92]
Shifted covering, square lattice60.347296...[92]
Staggered covering, square lattice60.376825(2)[92]
Random covering, square lattice60.367713(2)[92]
Parallel covering, triangular lattice100.237418...[92]
Staggered covering, triangular lattice100.237497(2)[92]
Random covering, triangular lattice100.235340(1)[92]

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.[93]

l (polymer length)zBond percolation
140.5(exact)[94]
240.47697(4)[94]
440.44892(6)[94]
840.41880(4)[94]

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

kzSite thresholdsBond thresholds
140.593(2)[95]0.5009(2)[95]
240.564(2)[95]0.4859(2)[95]
340.552(2)[95]0.4732(2)[95]
440.542(2)[95]0.4630(2)[95]
540.531(2)[95]0.4565(2)[95]
640.522(2)[95]0.4497(2)[95]
740.511(2)[95]0.4423(2)[95]
840.502(2)[95]0.4348(2)[95]
940.493(2)[95]0.4291(2)[95]
1040.488(2)[95]0.4232(2)[95]
1140.482(2)[95]0.4159(2)[95]
1240.476(2)[95]0.4114(2)[95]
1340.471(2)[95]0.4061(2)[95]
1440.467(2)[95]0.4011(2)[95]
1540.4011(2)[95]0.3979(2)[95]

Thresholds on 2D inhomogeneous lattices

LatticezSite percolation thresholdBond percolation threshold
bow-tie with p = 1⁄2 on one non-diagonal bond30.3819654(5),[96] (3 - \sqrt{5})/2[60]

Thresholds for 2D continuum models

SystemΦcηcnc
Disks of radius r0.67634831(2),[97] 0.6763475(6),[98] 0.676339(4),[99] 0.6764(4),[100] 0.6766(5),[101] 0.676(2),[102] 0.679,[103] 0.674[104] 0.676,[105] 0.680[106]1.1280867(5),[107] 1.1276(9),[108] 1.12808737(6),[97] 1.128085(2),[98] 1.128059(12),[99] 1.13,[citation needed] 0.8[109]1.43632505(10),[110] 1.43632545(8),[97] 1.436322(2),[98] 1.436289(16),[99] 1.436320(4),[111] 1.436323(3),[112] 1.438(2),[113] 1.216 (48)[114]
Disks of uniform radius (0,r)0.686610(7),[115] 0.6860(12),[100] 0.680[104]\rho_c r^2 = 1.108010(7)[115]
Ellipses, ε = 1.50.0043[103]0.004312.059081(7)[112]
Ellipses, ε = 5⁄30.65[116]1.05[116]2.28[116]
Ellipses, ε = 20.6287945(12),[112] 0.63[116]0.991000(3),[112] 0.99[116]2.523560(8),[112] 2.5[116]
Ellipses, ε = 30.56[116]0.82[116]3.157339(8),[112] 3.14[116]
Ellipses, ε = 40.5[116]0.69[116]3.569706(8),[112] 3.5[116]
Ellipses, ε = 50.455,[103] 0.455,[105] 0.46[116]0.607[103]3.861262(12),[112] 3.86[103]
Ellipses, ε = 64.079365(17)[112]
Ellipses, ε = 74.249132(16)[112]
Ellipses, ε = 84.385302(15)[112]
Ellipses, ε = 94.497000(8)[112]
Ellipses, ε = 100.301,[103] 0.303,[105] 0.30[116]0.358[103] 0.36[116]4.590416(23)[112] 4.56,[103] 4.5[116]
Ellipses, ε = 154.894752(30)[112]
Ellipses, ε = 200.178,[103] 0.17[116]0.196[103]5.062313(39),[112] 4.99[103]
Ellipses, ε = 500.081[103]0.084[103]5.393863(28),[112] 5.38[103]
Ellipses, ε = 1000.0417[103]0.0426[103]5.513464(40),[112] 5.42[103]
Ellipses, ε = 2000.021[116]0.0212[116]5.40[116]
Ellipses, ε = 10000.0043[103]0.004315.624756(22),[112] 5.5
Superellipses, ε = 1, m = 1.50.671[105]
Superellipses, ε = 2.5, m = 1.50.599[105]
Superellipses, ε = 5, m = 1.50.469[105]
Superellipses, ε = 10, m = 1.50.322[105]
disco-rectangles, ε = 1.51.894 [111]
disco-rectangles, ε = 22.245 [111]
Aligned squares of side \ell0.66675(2),[55] 0.66674349(3),[97] 0.66653(1),[117] 0.6666(4),[118] 0.668[104]1.09884280(9),[97] 1.0982(3),[117] 1.098(1)[118]1.09884280(9),[97] 1.0982(3),[117] 1.098(1)[118]
Randomly oriented squares0.62554075(4),[97] 0.6254(2)[118] 0.625,[105]0.9822723(1),[97] 0.9819(6)[118] 0.982278(14)[119]0.9822723(1),[97] 0.9819(6)[118] 0.982278(14)[119]
Randomly oriented squares within angle \pi/40.6255(1)[118]0.98216(15)[118]
Rectangles, ε = 1.10.624870(7)0.980484(19)1.078532(21)[119]
Rectangles, ε = 20.590635(5)0.893147(13)1.786294(26)[119]
Rectangles, ε = 30.5405983(34)0.777830(7)2.333491(22)[119]
Rectangles, ε = 40.4948145(38)0.682830(8)2.731318(30)[119]
Rectangles, ε = 50.4551398(31), 0.451[105]0.607226(6)3.036130(28)[119]
Rectangles, ε = 100.3233507(25), 0.319[105]0.3906022(37)3.906022(37)[119]
Rectangles, ε = 200.2048518(22)0.2292268(27)4.584535(54)[119]
Rectangles, ε = 500.09785513(36)0.1029802(4)5.149008(20)[119]
Rectangles, ε = 1000.0523676(6)0.0537886(6)5.378856(60)[119]
Rectangles, ε = 2000.02714526(34)0.02752050(35)5.504099(69)[119]
Rectangles, ε = 10000.00559424(6)0.00560995(6)5.609947(60)[119]
Sticks (needles) of length \ell5.637348(10),[120] 5.63726(2),[121] 5.6372858(6),[97] 5.637263(11),[119] 5.63724(18)[122]
3D sticks of length \ell and diameter d5.850923(14)[120]
sticks with log-normal length dist. STD=0.54.756(3)[122]
sticks with correlated angle dist. s=0.56.6076(4)[122]
Power-law disks, x = 2.050.993(1)[123]4.90(1)0.0380(6)
Power-law disks, x = 2.250.8591(5)[123]1.959(5)0.06930(12)
Power-law disks, x = 2.50.7836(4)[123]1.5307(17)0.09745(11)
Power-law disks, x = 40.69543(6)[123]1.18853(19)0.18916(3)
Power-law disks, x = 50.68643(13)[123]1.1597(3)0.22149(8)
Power-law disks, x = 60.68241(8)[123]1.1470(1)0.24340(5)
Power-law disks, x = 70.6803(8)[123]1.140(6)0.25933(16)
Power-law disks, x = 80.67917(9)[123]1.1368(5)0.27140(7)
Power-law disks, x = 90.67856(12)[123]1.1349(4)0.28098(9)
Voids around disks of radius r1 − Φc(disk) = 0.32355169(2),[97] 0.318(2),[124] 0.3261(6)[125]

For disks, n_c = 4 r^2 N / L^2 equals the critical number of disks per unit area, measured in units of the diameter 2r, where N is the number of objects and L is the system size

For disks, \eta_c = \pi r^2 N / L^2 = (\pi/4) n_c equals critical total disk area.

4 \eta_c gives the number of disk centers within the circle of influence (radius 2 r).

r_c = L \sqrt{\frac{\eta_c}{\pi N}} = \frac{L}{2} \sqrt{\frac{n_c}{N}} is the critical disk radius.

\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 \times 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.

For disks, Ref.[102] use \phi_c = 1 - e^{-\pi \times / 2} where x is the density of disks of radius 1/\sqrt{2}.

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 [112][116]

For more rectangle values, see [119]

Both ellipses and rectangles belong to the superellipses, with |x/a|^{2m}+|y/b|^{2m}=1. For more percolation values of superellipses, see.[105]

For the monodisperse particle systems, the percolation thresholds of concave-shaped superdisks are obtained as seen in [126]

For binary dispersions of disks, see [98][127][115]

Thresholds on 2D random and quasi-lattices

Latticez\overline zSite percolation thresholdBond percolation threshold
Relative neighborhood graph2.55760.796(2)[128]0.771(2)[128]
Voronoi tessellation30.71410(2),[129] 0.7151*[74]0.68,[130] 0.6670(1),[131] 0.6680(5),[132] 0.666931(5)[129]
Voronoi covering/medial40.666931(2)[129][131]0.53618(2)[129]
Randomized kagome/square-octagon, fraction r=1⁄240.6599[15]
Penrose rhomb dual40.6381(3)[71]0.5233(2)[71]
Gabriel graph40.6348(8),[133] 0.62[134]0.5167(6),[133] 0.52[134]
Random-line tessellation, dual40.586(2)[135]
Penrose rhomb40.5837(3),[71] 0.0.5610(6) (weighted bonds)[136] 0.58391(1)[137]0.483(5),[138] 0.4770(2)[71]
Octagonal lattice, "chemical" links (Ammann–Beenker tiling)40.585[139]0.48[139]
Octagonal lattice, "ferromagnetic" links5.170.543[139]0.40[139]
Dodecagonal lattice, "chemical" links3.630.628[139]0.54[139]
Dodecagonal lattice, "ferromagnetic" links4.270.617[139]0.495[139]
Delaunay triangulation61⁄2[140]0.3333(1)[131] 0.3326(5),[132] 0.333069(2)[129]
Uniform Infinite Planar Triangulation[141]61⁄2(2√3 – 1)/11 ≈ 0.2240[142][143]

*Theoretical estimate

Thresholds on 2D correlated systems

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

latticeαSite percolation thresholdBond percolation threshold
square30.561406(4)[144]
square20.550143(5)[144]
square0.10.508(4)[144]

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.

Latticehz\overline zSite percolation thresholdBond percolation threshold
simple cubic (open b.c.)2550.47424,[145] 0.4756[146]
bcc (open b.c.)20.4155[146]
hcp (open b.c.)20.2828[146]
diamond (open b.c.)20.5451[146]
simple cubic (open b.c.)30.4264[146]
bcc (open b.c.)30.3531[146]
bcc (periodic b.c.)30.21113018(38)[147]
hcp (open b.c.)30.2548[146]
diamond (open b.c.)30.5044[146]
simple cubic (open b.c.)40.3997,[145] 0.3998[146]
bcc (open b.c.)40.3232[146]
bcc (periodic b.c.)40.20235168(59)[147]
hcp (open b.c.)40.2405[146]
diamond (open b.c.)40.4842[146]
simple cubic (periodic b.c.)5660.278102(5)[147]
simple cubic (open b.c.)60.3708[146]
simple cubic (periodic b.c.)6660.272380(2)[147]
bcc (open b.c.)60.2948[146]
hcp (open b.c.)60.2261[146]
diamond (open b.c.)60.4642[146]
simple cubic (periodic b.c.)7660.3459514(12)[147]0.268459(1)[147]
simple cubic (open b.c.)80.3557,[145] 0.3565[146]
simple cubic (periodic b.c.)8660.265615(5)[147]
bcc (open b.c.)80.2811[146]
hcp (open b.c.)80.2190[146]
diamond (open b.c.)80.4549[146]
simple cubic (open b.c.)120.3411[146]
bcc (open b.c.)120.2688[146]
hcp (open b.c.)120.2117[146]
diamond (open b.c.)120.4456[146]
simple cubic (open b.c.)160.3219,[145] 0.3339[146]
bcc (open b.c.)160.2622[146]
hcp (open b.c.)160.2086[146]
diamond (open b.c.)160.4415[146]
simple cubic (open b.c.)320.3219,[145]
simple cubic (open b.c.)640.3165,[145]
simple cubic (open b.c.)1280.31398,[145]

Percolation in 3D

Latticez\overline zfilling factor*filling fraction*Site percolation thresholdBond percolation threshold
(10,3)-a oxide (or site-bond)[148]23 322.40.748713(22)[148]= (pc,bond(10,3) – a)1⁄2 = 0.742334(25)[149]
(10,3)-b oxide (or site-bond)[148]23 322.40.233[150]0.1740.745317(25)[148]= (pc,bond(10,3) – b)1⁄2 = 0.739388(22)[149]
silicon dioxide (diamond site-bond)[148]4,222 2⁄30.638683(35)[148]
Modified (10,3)-b[151]32,22 2⁄30.627[151]
(8,3)-a[149]330.577962(33)[149]0.555700(22)[149]
(10,3)-a[149] gyroid[152]330.571404(40)[149]0.551060(37)[149]
(10,3)-b[149]330.565442(40)[149]0.546694(33)[149]
cubic oxide (cubic site-bond)[148]6,233.50.524652(50)[148]
bcc dual40.4560(6)[153]0.4031(6)[153]
ice Ih44π √3 / 16 = 0.3400870.1470.433(11)[154]0.388(10)[155]
diamond (Ice Ic)44π √3 / 16 = 0.3400870.14623320.4299(8),[156] 0.4299870(4),[157] 0.426 ± 0.08,[158] 0.4297(4)[159] 0.4301(4),[160] 0.428(4),[161] 0.425(15),[162] 0.425,[43][50] 0.436(12)[154]0.3895892(5),[157] 0.3893(2),[160] 0.3893(3),[159] 0.388(5),[162] 0.3886(5),[156] 0.388(5)[161] 0.390(11)[155]
diamond dual6 2⁄30.3904(5)[153]0.2350(5)[153]
3D kagome (covering graph of the diamond lattice)6π √2 / 12 = 0.370240.14420.3895(2)[163] =pc(site) for diamond dual and pc(bond) for diamond lattice[153]0.2709(6)[153]
Bow-tie stack dual5 1⁄30.3480(4)[40]0.2853(4)[40]
honeycomb stack550.3701(2)[40]0.3093(2)[40]
octagonal stack dual550.3840(4)[40]0.3168(4)[40]
pentagonal stack5 1⁄30.3394(4)[40]0.2793(4)[40]
kagome stack660.4534500.15170.3346(4)[40]0.2563(2)[40]
fcc dual42,85 1⁄30.3341(5)[153]0.2703(3)[153]
simple cubic66π / 6 = 0.52359880.16315740.307(10),[162] 0.307,[43] 0.3115(5),[164] 0.3116077(2),[165] 0.311604(6),[166] 0.311605(5),[167] 0.311600(5),[168] 0.3116077(4),[169] 0.3116081(13),[170] 0.3116080(4),[171] 0.3116060(48),[172] 0.3116004(35),[173] 0.31160768(15)[157]0.247(5),[162] 0.2479(4),[156] 0.2488(2),[174] 0.24881182(10),[165] 0.2488125(25),[175] 0.2488126(5),[176]
hcp dual44,825 1⁄30.3101(5)[153]0.2573(3)[153]
dice stack5,86π √3 / 9 = 0.6046000.18130.2998(4)[40]0.2378(4)[40]
bow-tie stack770.2822(6)[40]0.2092(4)[40]
Stacked triangular / simple hexagonal880.26240(5),[177] 0.2625(2),[178] 0.2623(2)[40]0.18602(2),[177] 0.1859(2)[40]
octagonal (union-jack) stack6,1080.2524(6)[40]0.1752(2)[40]
bcc880.243(10),[162] 0.243,[43] 0.2459615(10),[171] 0.2460(3),[179] 0.2464(7),[156] 0.2458(2)[160]0.178(5),[162] 0.1795(3),[156] 0.18025(15),[174] 0.1802875(10)[176]
simple cubic with 3NN (same as bcc)880.2455(1),[180] 0.2457(7)[181]
fcc, D31212π / (3 √2) = 0.7404800.1475300.195,[43] 0.198(3),[182] 0.1998(6),[156] 0.1992365(10),[171] 0.19923517(20),[157] 0.1994(2),[160] 0.199236(4)[183]0.1198(3),[156] 0.1201635(10)[176] 0.120169(2)[183]
hcp1212π / (3 √2) = 0.7404800.1475450.195(5),[162] 0.1992555(10)[184]0.1201640(10),[184] 0.119(2)[162]
La2−x Srx Cu O412120.19927(2)[185]
simple cubic with 2NN (same as fcc)12120.1991(1)[180]
simple cubic with NN+4NN12120.15040(12),[186] 0.1503793(7)[52]0.1068263(7)[187]
simple cubic with 3NN+4NN14140.20490(12)[186]0.1012133(7)[187]
bcc NN+2NN (= sc(3,4) sc-3NN+4NN)14140.175,[43] 0.1686,(20)[188] 0.1759432(8)0.0991(5),[188] 0.1012133(7),[47] 0.1759432(8)[47]
Nanotube fibers on FCC14140.1533(13)[189]
simple cubic with NN+3NN14140.1420(1)[180]0.0920213(7)[187]
simple cubic with 2NN+4NN18180.15950(12)[186]0.0751589(9)[187]
simple cubic with NN+2NN18180.137,[50] 0.136,[190] 0.1372(1),[180] 0.13735(5),[citation needed] 0.1373045(5)[52]0.0752326(6)[187]
fcc with NN+2NN (=sc-2NN+4NN)18180.136,[43] 0.1361408(8)[52]0.0751589(9)[47]
simple cubic with short-length correlation6+6+0.126(1)[191]
simple cubic with NN+3NN+4NN20200.11920(12)[186]0.0624379(9)[187]
simple cubic with 2NN+3NN20200.1036(1)[180]0.0629283(7)[187]
simple cubic with NN+2NN+4NN24240.11440(12)[186]0.0533056(6)[187]
simple cubic with 2NN+3NN+4NN26260.11330(12)[186]0.0474609(9)
simple cubic with NN+2NN+3NN26260.097,[43] 0.0976(1),[180] 0.0976445(10), 0.0976444(6)[52]0.0497080(10)[187]
bcc with NN+2NN+3NN26260.095,[50] 0.0959084(6)[52]0.0492760(10)[47]
simple cubic with NN+2NN+3NN+4NN32320.10000(12),[186] 0.0801171(9)[52]0.0392312(8)[187]
fcc with NN+2NN+3NN42420.061,[50] 0.0610(5),[190] 0.0618842(8)[47]0.0290193(7)[47]
fcc with NN+2NN+3NN+4NN54540.0500(5)[190]
sc-1,2,3,4,5 simple cubic with NN+2NN+3NN+4NN+5NN56560.0461815(5)[52]0.0210977(7)[47]
sc-1,...,6 (2x2x2 cube [54])80800.0337049(9),[52] 0.03373(13)[54]0.0143950(10)[47]
sc-1,...,792920.0290800(10)[52]0.0123632(8)[47]
sc-1,...,81221220.0218686(6)[52]0.0091337(7)[47]
sc-1,...,91461460.0184060(10)[52]0.0075532(8)[47]
sc-1,...,101701700.0064352(8)[47]
sc-1,...,111781780.0061312(8)[47]
sc-1,...,122022020.0053670(10)[47]
sc-1,...,132502500.0042962(8)[47]
3x3x3 cube274274φc= 0.76564(1),[55] pc = 0.0098417(7),[55] 0.009854(6)[54]
4x4x4 cube636636φc=0.76362(1),[55] pc = 0.0042050(2),[55] 0.004217(3)[54]
5x5x5 cube12141250φc=0.76044(2),[55] pc = 0.0021885(2),[55] 0.002185(4)[54]
6x6x6 cube205620560.001289(2)[54]

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 [192]

Systempolymer Φc
percolating excluded volume of athermal polymer matrix (bond-fluctuation model on cubic lattice)0.4304(3)[193]

3D distorted lattices

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

Lattice\overline z\alphadSite percolation thresholdBond percolation threshold
cubic0.051.00.60254(3)[194]
0.11.006250.58688(4)[194]
0.151.0250.55075(2)[194]
0.1751.050.50645(5)[194]
0.21.10.44342(3)[194]

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=6k^2+18k-4

SystemkzSite coverage φcSite percolation threshold pc
1 × 2 dimer, cubic lattice2560.24542[54]0.045847(2)[54]
1 × 3 trimer, cubic lattice31040.19578[54]0.023919(9)[54]
1 × 4 stick, cubic lattice41640.16055[54]0.014478(7)[54]
1 × 5 stick, cubic lattice52360.13488[54]0.009613(8)[54]
1 × 6 stick, cubic lattice63200.11569[54]0.006807(2)[54]
2 × 2 plaquette, cubic lattice20.22710[54]0.021238(2)[54]
3 × 3 plaquette, cubic lattice30.18686[54]0.007632(5)[54]
4 × 4 plaquette, cubic lattice40.16159[54]0.003665(3)[54]
5 × 5 plaquette, cubic lattice50.14316[54]0.002058(5)[54]
6 × 6 plaquette, cubic lattice60.12900[54]0.001278(5)[54]

The coverage is calculated from p_c by \phi_c = 1-(1-p_c)^{3 k} for sticks, and \phi_c = 1-(1-p_c)^{3 k^2} for plaquettes.

Dimer percolation in 3D

SystemSite percolation thresholdBond percolation threshold
Simple cubic0.2555(1)[195]

Thresholds for 3D continuum models

All overlapping except for jammed spheres and polymer matrix.

SystemΦcηc
Spheres of radius r0.289,[196] 0.293,[197] 0.286,[198] 0.295.[104] 0.2895(5),[199] 0.28955(7),[200] 0.2896(7),[201] 0.289573(2),[202] 0.2896,[203] 0.2854,[204] 0.290,[205] 0.290,[206] 0.2895693(26)[207]0.3418(7),[199] 0.3438(13),[208] 0.341889(3),[202] 0.3360,[204] 0.34189(2)[117] [corrected], 0.341935(8),[209] 0.335,[210]
Oblate ellipsoids with major radius r and aspect ratio 4⁄30.2831[204]0.3328[204]
Prolate ellipsoids with minor radius r and aspect ratio 3⁄20.2757,[203] 0.2795,[204] 0.2763[205]0.3278[204]
Oblate ellipsoids with major radius r and aspect ratio 20.2537,[203] 0.2629,[204] 0.254[205]0.3050[204]
Prolate ellipsoids with minor radius r and aspect ratio 20.2537,[203] 0.2618,[204] 0.25(2),[211] 0.2507[205]0.3035,[204] 0.29(3)[211]
Oblate ellipsoids with major radius r and aspect ratio 30.2289[204]0.2599[204]
Prolate ellipsoids with minor radius r and aspect ratio 30.2033,[203] 0.2244,[204] 0.20(2)[211]0.2541,[204] 0.22(3)[211]
Oblate ellipsoids with major radius r and aspect ratio 40.2003[204]0.2235[204]
Prolate ellipsoids with minor radius r and aspect ratio 40.1901,[204] 0.16(2)[211]0.2108,[204] 0.17(3)[211]
Oblate ellipsoids with major radius r and aspect ratio 50.1757[204]0.1932[204]
Prolate ellipsoids with minor radius r and aspect ratio 50.1627,[204] 0.13(2)[211]0.1776,[204] 0.15(2)[211]
Oblate ellipsoids with major radius r and aspect ratio 100.0895,[203] 0.1058[204]0.1118[204]
Prolate ellipsoids with minor radius r and aspect ratio 100.0724,[203] 0.08703,[204] 0.07(2)[211]0.09105,[204] 0.07(2)[211]
Oblate ellipsoids with major radius r and aspect ratio 1000.01248[204]0.01256[204]
Prolate ellipsoids with minor radius r and aspect ratio 1000.006949[204]0.006973[204]
Oblate ellipsoids with major radius r and aspect ratio 10000.001275[204]0.001276[204]
Oblate ellipsoids with major radius r and aspect ratio 20000.000637[204]0.000637[204]
Spherocylinders with H/D = 10.2439(2)[201]
Spherocylinders with H/D = 40.1345(1)[201]
Spherocylinders with H/D = 100.06418(20)[201]
Spherocylinders with H/D = 500.01440(8)[201]
Spherocylinders with H/D = 1000.007156(50)[201]
Spherocylinders with H/D = 2000.003724(90)[201]
Aligned cylinders0.2819(2)[212]0.3312(1)[212]
Aligned cubes of side \ell = 2 a0.2773(2)[118] 0.27727(2),[55] 0.27730261(79)[172]0.3247(3),[117] 0.3248(3),[118] 0.32476(4)[212] 0.324766(1)[172]
Randomly oriented icosahedra0.3030(5)[213]
Randomly oriented dodecahedra0.2949(5)[213]
Randomly oriented octahedra0.2514(6)[213]
Randomly oriented cubes of side \ell = 2 a0.2168(2)[118] 0.2174,[203]0.2444(3),[118] 0.2443(5)[213]
Randomly oriented tetrahedra0.1701(7)[213]
Randomly oriented disks of radius r (in 3D)0.9614(5)[214]
Randomly oriented square plates of side \sqrt{\pi} r0.8647(6)[214]
Randomly oriented triangular plates of side \sqrt{2 \pi} /3^{1/4} r0.7295(6)[214]
Jammed spheres (average z = 6)0.183(3),[215] 0.1990,[216] see also contact network of jammed spheres below.0.59(1)[215] (volume fraction of all spheres)

\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.

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

For more results on void percolation around ellipsoids and elliptical plates, see.[217]

For more ellipsoid percolation values see.[204]

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.[201]

For superballs, m is the deformation parameter, the percolation values are given in.,[218][219] In addition, the thresholds of concave-shaped superballs are also determined in [126]

For cuboid-like particles (superellipsoids), m is the deformation parameter, more percolation values are given in.[203]

Void percolation in 3D

Void percolation refers to percolation in the space around overlapping objects. Here \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 \eta_c by \phi_c = e^{-\eta_c}. \eta_c is defined as in the continuum percolation section above.

SystemΦcηc
Voids around disks of radius r22.86(2)[217]
Voids around randomly oriented tetrahedra0.0605(6)[220]
Voids around oblate ellipsoids of major radius r and aspect ratio 320.5308(7)[221]0.6333[221]
Voids around oblate ellipsoids of major radius r and aspect ratio 160.3248(5)[221]1.125[221]
Voids around oblate ellipsoids of major radius r and aspect ratio 101.542(1)[217]
Voids around oblate ellipsoids of major radius r and aspect ratio 80.1615(4)[221]1.823[221]
Voids around oblate ellipsoids of major radius r and aspect ratio 40.0711(2)[221]2.643,[221] 2.618(5)[217]
Voids around oblate ellipsoids of major radius r and aspect ratio 23.239(4) [217]
Voids around prolate ellipsoids of aspect ratio 80.0415(7)[222]
Voids around prolate ellipsoids of aspect ratio 60.0397(7)[222]
Voids around prolate ellipsoids of aspect ratio 40.0376(7)[222]
Voids around prolate ellipsoids of aspect ratio 30.03503(50)[222]
Voids around prolate ellipsoids of aspect ratio 20.0323(5)[222]
Voids around aligned square prisms of aspect ratio 20.0379(5)[223]
Voids around randomly oriented square prisms of aspect ratio 200.0534(4)[223]
Voids around randomly oriented square prisms of aspect ratio 150.0535(4)[223]
Voids around randomly oriented square prisms of aspect ratio 100.0524(5)[223]
Voids around randomly oriented square prisms of aspect ratio 80.0523(6)[223]
Voids around randomly oriented square prisms of aspect ratio 70.0519(3)[223]
Voids around randomly oriented square prisms of aspect ratio 60.0519(5)[223]
Voids around randomly oriented square prisms of aspect ratio 50.0515(7)[223]
Voids around randomly oriented square prisms of aspect ratio 40.0505(7)[223]
Voids around randomly oriented square prisms of aspect ratio 30.0485(11)[223]
Voids around randomly oriented square prisms of aspect ratio 5/20.0483(8)[223]
Voids around randomly oriented square prisms of aspect ratio 20.0465(7)[223]
Voids around randomly oriented square prisms of aspect ratio 3/20.0461(14)[223]
Voids around hemispheres0.0455(6)[224]
Voids around aligned tetrahedra0.0605(6)[220]
Voids around randomly oriented tetrahedra0.0605(6)[220]
Voids around aligned cubes0.036(1),[55] 0.0381(3)[220]
Voids around randomly oriented cubes0.0452(6),[220] 0.0449(5)[223]
Voids around aligned octahedra0.0407(3)[220]
Voids around randomly oriented octahedra0.0398(5)[220]
Voids around aligned dodecahedra0.0356(3)[220]
Voids around randomly oriented dodecahedra0.0360(3)[220]
Voids around aligned icosahedra0.0346(3)[220]
Voids around randomly oriented icosahedra0.0336(7)[220]
Voids around spheres0.034(7),[225] 0.032(4),[226] 0.030(2),[124] 0.0301(3),[227] 0.0294,[222] 0.0300(3),[228] 0.0317(4),[229] 0.0308(5)[224] 0.0301(1),[221] 0.0301(1)[220]3.506(8),[228] 3.515(6),[217] 3.510(2)[108]

Thresholds on 3D random and quasi-lattices

Latticez\overline zSite percolation thresholdBond percolation threshold
Contact network of packed spheres60.310(5),[215] 0.287(50),[230] 0.3116(3),[216]
Random-plane tessellation, dual60.290(7)[231]
Icosahedral Penrose60.285[232]0.225[232]
Penrose w/2 diagonals6.7640.271[232]0.207[232]
Penrose w/8 diagonals12.7640.188[232]0.111[232]
Voronoi network15.540.1453(20)[188]0.0822(50)[188]

Thresholds for other 3D models

Latticez\overline zSite percolation thresholdCritical coverage fraction \phi_cBond percolation threshold
Drilling percolation, simple cubic lattice*660.6345(3),[233] 0.6339(5),[234] 0.633965(15)[235]0.25480
Drill in z direction on cubic lattice, remove single sites660.592746 (columns), 0.4695(10) (sites)[236]0.2784
Random tube model, simple cubic lattice†0.231456(6)[237]
Pac-Man percolation, simple cubic lattice0.139(6)[238]

^* In drilling percolation, the site threshold p_c represents the fraction of columns in each direction that have not been removed, and \phi_c=p_c^3. For the 1d drilling, we have \phi_c = p_c(columns) p_c(sites).

† In tube percolation, the bond threshold represents the value of the parameter \mu such that the probability of putting a bond between neighboring vertical tube segments is 1-e^{-\mu h_i}, where h_i is the overlap height of two adjacent tube segments.[237]

Thresholds in different dimensional spaces

Continuum models in higher dimensions

dSystemΦcηc
4Overlapping hyperspheres0.1223(4)[117]0.1300(13),[208] 0.1304(5),[117] 0.1210268(19)[207]
4Aligned hypercubes0.1132(5),[117] 0.1132348(17)[172]0.1201(6)[117]
4Voids around hyperspheres0.00211(2)[125]6.161(10)[125] 6.248(2),[108]
5Overlapping hyperspheres0.0544(6),[208] 0.05443(7),[117] 0.0522524(69)[207]
5Aligned hypercubes0.04900(7),[117] 0.0481621(13)[172]0.05024(7)[117]
5Voids around hyperspheres1.26(6)×10−4[125]8.98(4),[125] 9.170(8)[108]
6Overlapping hyperspheres0.02391(31),[208] 0.02339(5)[117]
6Aligned hypercubes0.02082(8),[117] 0.0213479(10)[172]0.02104(8)[117]
6Voids around hyperspheres8.0(6)×10−6[125]11.74(8),[125] 12.24(2),[108]
7Overlapping hyperspheres0.01102(16),[208] 0.01051(3)[117]
7Aligned hypercubes0.00999(5),[117] 0.0097754(31)[172]0.01004(5)[117]
7Voids around hyperspheres15.46(5)[108]
8Overlapping hyperspheres0.00516(8),[208] 0.004904(6)[117]
8Aligned hypercubes0.004498(5)[117]
8Voids around hyperspheres18.64(8)[108]
9Overlapping hyperspheres0.002353(4)[117]
9Aligned hypercubes0.002166(4)[117]
9Voids around hyperspheres22.1(4)[108]
10Overlapping hyperspheres0.001138(3)[117]
10Aligned hypercubes0.001058(4)[117]
11Overlapping hyperspheres0.0005530(3)[117]
11Aligned hypercubes0.0005160(3)[117]

\eta_c = (\pi^{d/2}/ \Gamma[d/2 + 1]) r^d N / L^d.

In 4d, \eta_c = (1/2) \pi^2 r^4 N / L^4.

In 5d, \eta_c = (8/15) \pi^2 r^5 N / L^5.

In 6d, \eta_c = (1/6) \pi^3 r^6 N / L^6.

\phi_c = 1 - e^{-\eta_c} is the critical volume fraction, valid for overlapping objects.

For void models, \phi_c = e^{-\eta_c} is the critical void fraction, and \eta_c is the total volume of the overlapping objects

Thresholds on hypercubic lattices

dzSite thresholdsBond thresholds
480.198(1)[239] 0.197(6),[240] 0.1968861(14),[241] 0.196889(3),[242] 0.196901(5),[243] 0.19680(23),[244] 0.1968904(65),[172] 0.19688561(3)[245]0.1600(1),[246] 0.16005(15),[174] 0.1601314(13),[241] 0.160130(3),[242] 0.1601310(10),[175] 0.1601312(2),[247] 0.16013122(6)[245]
5100.141(1),0.198(1)[239] 0.141(3),[240] 0.1407966(15),[241] 0.1407966(26),[172] 0.14079633(4)[245]0.1181(1),[246] 0.118(1),[248] 0.11819(4),[174] 0.118172(1),[241] 0.1181718(3)[175] 0.11817145(3)[245]
6120.106(1),[239] 0.108(3),[240] 0.109017(2),[241] 0.1090117(30),[172] 0.109016661(8)[245]0.0943(1),[246] 0.0942(1),[249] 0.0942019(6),[241] 0.09420165(2)[245]
7140.05950(5),[249] 0.088939(20),[250] 0.0889511(9),[241] 0.0889511(90),[172] 0.088951121(1),[245]0.0787(1),[246] 0.078685(30),[249] 0.0786752(3),[241] 0.078675230(2)[245]
8160.0752101(5),[241] 0.075210128(1)[245]0.06770(5),[249] 0.06770839(7),[241] 0.0677084181(3)[245]
9180.0652095(3),[241] 0.0652095348(6)[245]0.05950(5),[249] 0.05949601(5),[241] 0.0594960034(1)[245]
10200.0575930(1),[241] 0.0575929488(4)[245]0.05309258(4),[241] 0.0530925842(2)[245]
11220.05158971(8),[241] 0.0515896843(2)[245]0.04794969(1),[241] 0.04794968373(8)[245]
12240.04673099(6),[241] 0.0467309755(1)[245]0.04372386(1),[241] 0.04372385825(10)[245]
13260.04271508(8),[241] 0.04271507960(10)[245]0.04018762(1),[241] 0.04018761703(6)[245]

For thresholds on high dimensional hypercubic lattices, we have the asymptotic series expansions [240] [248] [251]

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^\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 \sigma = 2 d - 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

dlatticezSite thresholdsBond thresholds
4diamond50.2978(2)[160]0.2715(3)[160]
4kagome80.2715(3)[163]0.177(1)[160]
4bcc160.1037(3)[160]0.074(1),[160] 0.074212(1)[247]
4fcc, D4, hypercubic 2NN240.0842(3),[160] 0.08410(23),[244] 0.0842001(11)[183]0.049(1),[160] 0.049517(1),[247] 0.0495193(8)[183]
4hypercubic NN+2NN320.06190(23),[244] 0.0617731(19)[252]0.035827(1),[247] 0.0338047(27)[252]
4hypercubic 3NN320.04540(23)[244]
4hypercubic NN+3NN400.04000(23)[244]0.0271892(22)[252]
4hypercubic 2NN+3NN560.03310(23)[244]0.0194075(15)[252]
4hypercubic NN+2NN+3NN640.03190(23),[244] 0.0319407(13)[252]0.0171036(11)[252]
4hypercubic NN+2NN+3NN+4NN880.0231538(12)[252]0.0122088(8)[252]
4hypercubic NN+...+5NN1360.0147918(12)[252]0.0077389(9)[252]
4hypercubic NN+...+6NN2320.0088400(10)[252]0.0044656(11)[252]
4hypercubic NN+...+7NN2960.0070006(6)[252]0.0034812(7)[252]
4hypercubic NN+...+8NN3200.0064681(9)[252]0.0032143(8)[252]
4hypercubic NN+...+9NN4240.0048301(9)[252]0.0024117(7)[252]
5diamond60.2252(3)[160]0.2084(4)[163]
5kagome100.2084(4)[163]0.130(2)[160]
5bcc320.0446(4)[160]0.033(1)[160]
5fcc, D5, hypercubic 2NN400.0431(3),[160] 0.0435913(6)[183]0.026(2),[160] 0.0271813(2)[183]
5hypercubic NN+2NN500.0334(2)[253]0.0213(1)[253]
6diamond70.1799(5)[160]0.1677(7)[163]
6kagome120.1677(7)[163]
6fcc, D6600.0252(5),[160] 0.02602674(12)[183]0.01741556(5)[183]
6bcc640.0199(5)[160]
6E6[183]720.02194021(14)[183]0.01443205(8)[183]
7fcc, D7840.01716730(5)[183]0.012217868(13)[183]
7E7[183]1260.01162306(4)[183]0.00808368(2)[183]
8fcc, D81120.01215392(4)[183]0.009081804(6)[183]
8E8[183]2400.00576991(2)[183]0.004202070(2)[183]
9fcc, D91440.00905870(2)[183]0.007028457(3)[183]
9\Lambda_9[183]2720.00480839(2)[183]0.0037006865(11)[183]
10fcc, D101800.007016353(9)[183]0.005605579(6)[183]
11fcc, D112200.005597592(4)[183]0.004577155(3)[183]
12fcc, D122640.004571339(4)[183]0.003808960(2)[183]
13fcc, D133120.003804565(3)[183]0.0032197013(14)[183]

Thresholds in one-dimensional long-range percolation

In a one-dimensional chain we establish bonds between distinct sites i and j with probability p=\frac{C}{|i-j|^{1+\sigma}} decaying as a power-law with an exponent \sigma>0. Percolation occurs[254][255] at a critical value C_c<1 for \sigma<1. The numerically determined percolation thresholds are given by:[256]

\sigmaC_cCritical thresholds C_c as a function of \sigma.[256]
The dotted line is the rigorous lower bound.[254]
0.10.047685(8)
0.20.093211(16)
0.30.140546(17)
0.40.193471(15)
0.50.25482(5)
0.60.327098(6)
0.70.413752(14)
0.80.521001(14)
0.90.66408(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.

Latticez\overline zSite percolation thresholdBond percolation threshold
LowerUpperLowerUpper
{3,7} hyperbolic770.26931171(7),[257] 0.20[258]0.73068829(7),[257] 0.73(2)[258]0.20,[259] 0.1993505(5)[257]0.37,[259] 0.4694754(8)[257]
{3,8} hyperbolic880.20878618(9)[257]0.79121382(9)[257]0.1601555(2)[257]0.4863559(6)[257]
{3,9} hyperbolic990.1715770(1)[257]0.8284230(1)[257]0.1355661(4)[257]0.4932908(1)[257]
{4,5} hyperbolic550.29890539(6)[257]0.8266384(5)[257]0.27,[259] 0.2689195(3)[257]0.52,[259] 0.6487772(3)[257]
{4,6} hyperbolic660.22330172(3)[257]0.87290362(7)[257]0.20714787(9)[257]0.6610951(2)[257]
{4,7} hyperbolic770.17979594(1)[257]0.89897645(3)[257]0.17004767(3)[257]0.66473420(4)[257]
{4,8} hyperbolic880.151035321(9)[257]0.91607962(7)[257]0.14467876(3)[257]0.66597370(3)[257]
{4,9} hyperbolic880.13045681(3)[257]0.92820305(3)[257]0.1260724(1)[257]0.66641596(2)[257]
{5,5} hyperbolic550.26186660(5)[257]0.89883342(7)[257]0.263(10),[260] 0.25416087(3)[257]0.749(10)[260] 0.74583913(3)[257]
{7,3} hyperbolic330.54710885(10)[257]0.8550371(5),[257] 0.86(2)[258]0.53,[259] 0.551(10),[260] 0.5305246(8)[257]0.72,[259] 0.810(10),[260] 0.8006495(5)[257]
{∞,3} Cayley tree331⁄21⁄2[259]1[259]
Enhanced binary tree (EBT)0.304(1),[261] 0.306(10),[260] (√13 − 3)/2 = 0.302776[262]0.48,[259] 0.564(1),[261] 0.564(10),[260] 1⁄2[262]
Enhanced binary tree dual0.436(1),[261] 0.452(10)[260]0.696(1),[261] 0.699(10)[260]
Non-Planar Hanoi Network (HN-NP)0.319445[263]0.381996[263]
Cayley tree with grandparents80.158656326[264]

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,\ell}(P,Q) + p_{c,u}(Q,P) = 1. For site percolation, 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 )

Thresholds for directed percolation

LatticezSite percolation thresholdBond percolation threshold
(1+1)-d honeycomb1.50.8399316(2),[265] 0.839933(5),[266] = \sqrt{p_c({\mathrm{site}})} of (1+1)-d sq.0.8228569(2),[265] 0.82285680(6)[265]
(1+1)-d kagome20.7369317(2),[265] 0.73693182(4)[267]0.6589689(2),[265] 0.65896910(8)[265]
(1+1)-d square, diagonal20.705489(4),[268] 0.705489(4),[269] 0.70548522(4),[270] 0.70548515(20),[267] 0.7054852(3),[265]0.644701(2),[271] 0.644701(1),[272] 0.644701(1),[268] 0.6447006(10),[266] 0.64470015(5),[273] 0.644700185(5),[270] 0.6447001(2),[265] 0.643(2)[274]
(1+1)-d triangular30.595646(3),[268] 0.5956468(5),[273] 0.5956470(3)[265]0.478018(2),[268] 0.478025(1),[273] 0.4780250(4)[265] 0.479(3)[274]
(2+1)-d simple cubic, diagonal planes30.43531(1),[275] 0.43531411(10)[265]0.382223(7),[275] 0.38222462(6)[265] 0.383(3)[274]
(2+1)-d square nn (= bcc)40.3445736(3),[276] 0.344575(15)[277] 0.3445740(2)[265]0.2873383(1),[278] 0.287338(3)[275] 0.28733838(4)[265] 0.287(3)[274]
(2+1)-d fcc0.199(2))[274]
(3+1)-d hypercubic, diagonal40.3025(10),[279] 0.30339538(5)[265]0.26835628(5),[265] 0.2682(2)[274]
(3+1)-d cubic, nn60.2081040(4)[276]0.1774970(5)[175]
(3+1)-d bcc80.160950(30),[277] 0.16096128(3)[265]0.13237417(2)[265]
(4+1)-d hypercubic, diagonal50.23104686(3)[265]0.20791816(2),[265] 0.2085(2)[274]
(4+1)-d hypercubic, nn80.1461593(2),[276] 0.1461582(3)[280]0.1288557(5)[175]
(4+1)-d bcc160.075582(17),[277] 0.0755850(3),[280] 0.07558515(1)[265]0.063763395(5)[265]
(5+1)-d hypercubic, diagonal60.18651358(2)[265]0.170615155(5),[265] 0.1714(1)[274]
(5+1)-d hypercubic, nn100.1123373(2)[276]0.1016796(5)[175]
(5+1)-d hypercubic bcc320.035967(23),[277] 0.035972540(3)[265]0.0314566318(5)[265]
(6+1)-d hypercubic, diagonal70.15654718(1)[265]0.145089946(3),[265] 0.1458[274]
(6+1)-d hypercubic, nn120.0913087(2)[276]0.0841997(14)[175]
(6+1)-d hypercubic bcc640.017333051(2)[265]0.01565938296(10)[265]
(7+1)-d hypercubic, diagonal80.135004176(10)[265]0.126387509(3),[265] 0.1270(1)[274]
(7+1)-d hypercubic,nn140.07699336(7)[276]0.07195(5)[175]
(7+1)-d bcc1280.008 432 989(2)[265]0.007 818 371 82(6)[265]

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

LatticezSite percolation thresholdBond percolation threshold
(1+1)-d square30.4395(3),[281]
(1+1)-d square50.2249(3)[281]
(1+1)-d square70.1470(2)[281]
(1+1)-d square90.1081(2)[281]
(1+1)-d square110.0851(2)[281]
(1+1)-d square130.0701(2)[281]
(1+1)-d tilted sq20.6447(2)[281]
(1+1)-d tilted sq40.3272(2)[281]
(1+1)-d tilted sq60.2121(3)[281]
(1+1)-d tilted sq80.1553(3)[281]
(1+1)-d tilted sq100.1220(2)[281]
(1+1)-d tilted sq120.0999(2)[281]

For large z, pc ~ 1/z [281]

Site-Bond Directed Percolation

pb = bond threshold

ps = site threshold

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

P0 = probability that no sites are connected = (1-ps) + ps(1-pb)2

P2 = probability that exactly one descendant is connected to the upper vertex (two connected together) = ps pb (1-pb)

P3 = probability that both descendants are connected to the original vertex (all three connected together)= ps pb2

Normalization: P0 + 2P2 + P3 = 1

LatticezpspbP0P2P3
(1+1)-d square [282]30.64470110.1262370.2290620.415639
0.70.935850.1483760.1965290.458567
0.750.885650.1697030.1660590.498178
0.80.841350.1923040.1346160.538464
0.850.801900.2161430.1022420.579373
0.90.766450.2412150.0689810.620825
0.950.734500.2673360.0348890.662886
10.7054890.29451100.705489

Isotropic/Directed Percolation

Here we have a cross between ordinary bond percolation (OP) and directed percolation (DP)[283]. On an oriented system such as shown in the figure "(1+1)d Square Lattice" above, we consider the down probability p↓ = p pd and the up probability p↑ = p(1 − pd ), with p representing the average bond occupation probability and pd controlling the anisotropy. When pd = 0 or 1, we have pure DP, while when pd = 1/2 we have the random diode model or essentially OP, with the threshold twice the OP value. For other values of pd, we have a mixture of the two types of percolation. For a given pd, the critical values of p = pc are given below:

Latticedzpdpcp↓p↑
(1+1)-d DP2210.644700185(5)[270]0.644700185(5)0
diagonal square [283]240.80.768708(1)0.6149660.153742
diagonal square [283]240.60.929668(3)0.557801s0.371867
2d ordinary perc.240.51.00.50.5
(2+1)-d diagonal DP3310.38222462(6)[265]0.382224620
diagonal cubic360.80.430941(2)[283]0.344752820.086188
diagonal cubic360.60.481310(2)[283]0.2887860.192524
3d Ordinary perc.360.50.49762364(20)[165]0.248811820.24881182

Exact critical manifolds of inhomogeneous systems

Inhomogeneous triangular lattice bond percolation[19]

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

Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation[19]

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[6][284]

1 - 3(s_1s_2)^2 + (s_1s_2)^3 = 0, or 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[285]

p_3 = 1 - p_1; \qquad p_4 = 1 - p_2

Inhomogeneous martini lattice, bond percolation[78][286]

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 martini lattice with outside enclosing triangle of bonds, probabilities y, x, z from inside to outside, bond percolation[286]

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

Inhomogeneous checkerboard lattice, bond percolation[61][96]

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[60][96]

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.

Rigidity percolation

Assuming a finite graph with unbending bonds, rigidity percolation refers to a situation where the entire graph is rigid everywhere with respect to shear forces being put on it.[287][288][289][290] Another way to say this is that constraints are sufficient to eliminate all zero-frequency vibrational modes, transforming a mechanically floppy network into one capable of supporting stress[291].

The Geiringer–Laman theorem gives a combinatorial characterization of generically rigid graphs in 2-dimensional Euclidean space

Generic lattices have bonds of different lengths, and can be made by randomly displacing the sites of a regular lattice.

Numerical results:

Feng, Sen (1984)[292] pc = 0.58, f = 2.4 ± 0.4.

Lemieux, Breton, Tremblay (1985)[293] pc= 0.649, f = 1.4

Sahimi, Goddard (1985) bond triangular pc=0.65[294]

Arababi, Sahimi (1988): [295]. 3d bond cubic elastic network pc = 0.2492,

Roux, Hansen (1988): central force elastic network: p* = 0.642(2), flv = 3.0(4) , glv = 0.97(2)[296]

Jacobs and Thorpe (1996)[297] Bond threshold, triangular lattice: pc = 0.6602(3), pc = 0.69755(3)

Site percolation, triangular lattice pc = 0.69755(3),[297] 0.6975(3)[298]

Correlation-length exponent: ν = 1.16(3), 1.19(1),[299] 1.21(6)[297], 1/ν = 0.850(3)[291]

Arbibi Sahimi (1993)[300]: 2d bond tri: pc = 0.641(1), site: pc = 0.713(2).

Moukarzel and Duxbury (1995)[298]: 0.6975(3) 𝛼 = -0.48(5) β = 0.175(2)

Fractal dimension df = 1.86(2)[297]. 1.853(5)[301], 1.850(2)[291]

Backbone fractal dimension db = 1.80(3),[297] 1.78(2)[298]

Duxbury, Jacobs, Thorpe, Moukarzel (1999)[302] Bethe lattice z = 6, pc = 0.656

Chubynsky and Thorpe (2007)[303]. 3d: bond fcc, pc = 0.495. bcc: pc = 0.7485

Javerzat (2024) [304] 2d hull fractal dimension: df =1.355(10).

A fast algorithm for 2D Rigidity Percolation, Javerzat and Notarmuzi.[305] (2026) ν = 1.1694(8), df = 1.8423(7), pc = 0.6602741(4).

Lu, Song, Shi, Li, Deng (2026)[306] pc = 0.6602778(10), 1/ν = 0.850(3), and df = 1.850(2),

See also

References

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