Percolation threshold

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 p_c, 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 p_c. For finite large systems, P(p_c) is a constant whose value depends upon the shape of the system; for the square system discussed above, P(p_c)=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, n_s(p_c) ~ 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.

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

${\displaystyle p_{c}={\frac {1}{g_{1}'(1)}}={\frac {\langle k\rangle }{\langle k^{2}\rangle -\langle k\rangle }}}$.

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

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

${\displaystyle p_{c}={\frac {1}{1-C}}{\frac {1}{g_{1}'(1)}}.}$

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

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
3-12 or super-kagome, (3, 122 )	3	3	0.807900764... = (1 − 2 sin (π/18))1⁄2[7]	0.74042195(80),[8] 0.74042077(2),[9] 0.740420800(2),[10] 0.7404207988509(8),[11][12] 0.740420798850811610(2),[13]
cross, truncated trihexagonal (4, 6, 12)	3	3	0.746,[14] 0.750,[15] 0.747806(4),[7] 0.7478008(2)[11]	0.6937314(1),[11] 0.69373383(72),[8] 0.693733124922(2)[13]
square octagon, bathroom tile, 4-8, truncated square (4, 82)	3	-	0.729,[14] 0.729724(3),[7] 0.7297232(5)[11]	0.6768,[16] 0.67680232(63),[8] 0.6768031269(6),[11] 0.6768031243900113(3),[13]
honeycomb (63)	3	3	0.6962(6),[17] 0.697040230(5),[11] 0.6970402(1),[18] 0.6970413(10),[19] 0.697043(3),[7]	0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=0[20]
kagome (3, 6, 3, 6)	4	4	0.652703645... = 1 − 2 sin(π/18)[20]	0.5244053(3),[21] 0.52440516(10),[19] 0.52440499(2),[18] 0.524404978(5),[9] 0.52440572...,[22] 0.52440500(1),[10] 0.524404999173(3),[11][12] 0.524404999167439(4)[23] 0.52440499916744820(1)[13]
ruby,[24] rhombitrihexagonal (3, 4, 6, 4)	4	4	0.620,[14] 0.621819(3),[7] 0.62181207(7)[11]	0.52483258(53),[8] 0.5248311(1),[11] 0.524831461573(1)[13]
square (44)	4	4	0.59274(10),[25] 0.59274605079210(2),[23] 0.59274601(2),[11] 0.59274605095(15),[26] 0.59274621(13),[27] 0.592746050786(3),[28] 0.5927460507896(1),[29] 0.59274621(33),[30] 0.59274598(4),[31][32] 0.59274605(3),[18] 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)	5	5	0.579[15] 0.579498(3)[7]	0.43430621(50),[8] 0.43432764(3),[11] 0.4343283172240(6),[13]
snub square, puzzle (32, 4, 3, 4 )	5	5	0.550,[14][38] 0.550806(3)[7]	0.41413743(46),[8] 0.4141378476(7),[11] 0.4141378565917(1),[13]
frieze, elongated triangular(33, 42)	5	5	0.549,[14] 0.550213(3),[7] 0.5502(8)[39]	0.4196(6),[39] 0.41964191(43),[8] 0.41964044(1),[11] 0.41964035886369(2)[13]
triangular (36)	6	6	1⁄2	0.347296355... = 2 sin (π/18), 1 + p3 − 3p = 0[20]

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.

In this section, sq-1,2,3 corresponds to square (NN+2NN+3NN),^[40] etc. Equivalent to square-2N+3N+4N,^[41] sq(1,2,3).^[42] tri = triangular, hc = honeycomb.

Lattice	z	Site percolation threshold	Bond percolation threshold
sq-1, sq-2, sq-3, sq-5	4	0.5927...[40][41] (square site)
sq-1,2, sq-2,3, sq-3,5... 3x3 square	8	0.407...[40][41][43] (square matching)	0.25036834(6),[18] 0.2503685,[44] 0.25036840(4)[45]
sq-1,3	8	0.337[40][41]	0.2214995[44]
sq-2,5: 2NN+5NN	8	0.337[41]
hc-1,2,3: honeycomb-NN+2NN+3NN	12	0.300,[42] 0.300,[15] 0.302960... = 1-pc(site, hc)[46]
tri-1,2: triangular-NN+2NN	12	0.295,[42] 0.289,[15] 0.290258(19)[47]
tri-2,3: triangular-2NN+3NN	12	0.232020(36),[48] 0.232020(20)[47]
sq-4: square-4NN	8	0.270...[41]
sq-1,5: square-NN+5NN (r ≤ 2)	8	0.277[41]
sq-1,2,3: square-NN+2NN+3NN	12	0.292,[49] 0.290(5)[50] 0.289,[15] 0.288,[40][41] 0.2891226(14)[51]	0.1522203[44]
sq-2,3,5: square-2NN+3NN+5NN	12	0.288[41]
sq-1,4: square-NN+4NN	12	0.236[41]
sq-2,4: square-2NN+4NN	12	0.225[41]
tri-4: triangular-4NN	12	0.192450(36),[48] 0.1924428(50)[47]
hc-2,4: honeycomb-2NN+4NN	12	0.2374[52]
tri-1,3: triangular-NN+3NN	12	0.264539(21)[47]
tri-1,2,3: triangular-NN+2NN+3NN	18	0.225,[49] 0.215,[15] 0.215459(36)[48] 0.2154657(17)[47]
sq-3,4: 3NN+4NN	12	0.221[41]
sq-1,2,5: NN+2NN+5NN	12	0.240[41]	0.13805374[44]
sq-1,3,5: NN+3NN+5NN	12	0.233[41]
sq-4,5: 4NN+5NN	12	0.199[41]
sq-1,2,4: NN+2NN+4NN	16	0.219[41]
sq-1,3,4: NN+3NN+4NN	16	0.208[41]
sq-2,3,4: 2NN+3NN+4NN	16	0.202[41]
sq-1,4,5: NN+4NN+5NN	16	0.187[41]
sq-2,4,5: 2NN+4NN+5NN	16	0.182[41]
sq-3,4,5: 3NN+4NN+5NN	16	0.179[41]
sq-1,2,3,5 asterisk pattern	16	0.208[41]	0.1032177[44]
tri-4,5: 4NN+5NN	18	0.140250(36),[48]
sq-1,2,3,4: NN+2NN+3NN+4NN (${\displaystyle r\leq {\sqrt {5}}}$)	20	0.19671(9),[53] 0.196,[41] 0.196724(10),[54] 0.1967293(7)[51]	0.0841509[44]
sq-1,2,4,5: NN+2NN+4NN+5NN	20	0.177[41]
sq-1,3,4,5: NN+3NN+4NN+5NN	20	0.172[41]
sq-2,3,4,5: 2NN+3NN+4NN+5NN	20	0.167[41]
sq-1,2,3,5,6 asterisk pattern	20		0.0783110[44]
sq-1,2,3,4,5: NN+2NN+3NN+4NN+5NN (${\displaystyle r\leq {\sqrt {8}}}$, also within a 5 x 5 square)	24	0.164,[15] 0.164,[41] 0.1647124(6)[51]
sq-1,2,3,4,6: NN+2NN+3NN+4NN+6NN (diamond ${\displaystyle r\leq 3}$)	24	0.16134,[55]
tri-1,4,5: NN+4NN+5NN	24	0.131660(36)[48]
sq-1,...,6: NN+...+6NN (r≤3)	28	0.142,[15] 0.1432551(9)[51]	0.0558493[44]
tri-2,3,4,5: 2NN+3NN+4NN+5NN	30	0.117460(36)[48] 0.135823(27)[47]
tri-1,2,3,4,5: NN+2NN+3NN+4NN+5NN	36	0.115,[15] 0.115740(36),[48] 0.1157399(58)[47]
sq-1,...,7: NN+...+7NN (${\displaystyle r\leq {\sqrt {10}}}$)	36	0.113,[15] 0.1153481(9)[51]	0.04169608[44]
sq lat, diamond boundary: dist. ≤ 4	40	0.105(5)[50]
sq-1,...,8: NN+..+8NN (${\displaystyle r\leq {\sqrt {13}}}$)	44	0.095,[38] 0.095765(5),[54] 0.09580(2),[53] 0.0957661(9)[51]
sq-1,...,9: NN+..+9NN (r≤4)	48	0.086[15]	0.02974268[44]
sq-1,...,11: NN+...+11NN (${\displaystyle r\leq {\sqrt {18}}}$)	60		0.02301190(3)[44]
sq-1,...,23 (r ≤ 7)	148		0.008342595[45]
sq-1,...,32: NN+...+32NN (${\displaystyle r\leq {\sqrt {72}}}$)	224		0.0053050415(33)[44]
sq-1,...,86: NN+...+86NN (r≤15)	708		0.001557644(4)[56]
sq-1,...,141: NN+...+141NN (${\displaystyle r\leq {\sqrt {389}}}$)	1224		0.000880188(90)[44]
sq-1,...,185: NN+...+185NN (r≤23)	1652		0.000645458(4)[56]
sq-1,...,317: NN+...+317NN (r≤31)	3000		0.000349601(3)[56]
sq-1,...,413: NN+...+413NN (${\displaystyle r\leq {\sqrt {1280}}}$)	4016		0.0002594722(11)[44]
sq lat, diamond boundary: dist. ≤ 6	84	0.049(5)[50]
sq lat, diamond boundary: dist. ≤ 8	144	0.028(5)[50]
sq lat, diamond boundary: dist. ≤ 10	220	0.019(5)[50]
2x2 touching lattice squares* (same as sq-1,2,3,4)	20	φc = 0.58365(2),[54] pc = 0.196724(10),[54] 0.19671(9),[53]
3x3 touching lattice squares* (same as sq-1,...,8))	44	φc = 0.59586(2),[54] pc = 0.095765(5),[54] 0.09580(2)[53]
4x4 touching lattice squares*	76	φc = 0.60648(1),[54] pc = 0.0566227(15),[54] 0.05665(3),[53]
5x5 touching lattice squares*	116	φc = 0.61467(2),[54] pc = 0.037428(2),[54] 0.03745(2),[53]
6x6 touching lattice squares*	220	pc = 0.02663(1),[53]
10x10 touching lattice squares*	436	φc = 0.63609(2),[54] pc = 0.0100576(5)[54]
within 11 x 11 square (r=5)	120		0.01048079(6)[56]
within 15 x 15 square (r=7)	224		0.005287692(22)[56]
20x20 touching lattice squares*	1676	φc = 0.65006(2),[54] pc = 0.0026215(3)[54]
within 31 x 31 square (r=15)	960		0.001131082(5)[56]
100x100 touching lattice squares*	40396	φc = 0.66318(2),[54] pc = 0.000108815(12)[54]
1000x1000 touching lattice squares*	4003996	φc = 0.66639(1),[54] pc = 1.09778(6)E-06 [54]

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.^[40]

• For overlapping or touching squares, ${\displaystyle p_{c}}$(site) given here is the net fraction of sites occupied ${\displaystyle \phi _{c}}$ similar to the ${\displaystyle \phi _{c}}$ in continuum percolation. The case of a 2×2 square is equivalent to percolation of a square lattice NN+2NN+3NN+4NN or sq-1,2,3,4 with threshold ${\displaystyle 1-(1-\phi _{c})^{1/4}=0.196724(10)\ldots }$ with ${\displaystyle \phi _{c}=0.58365(2)}$.^[54] The 3×3 square corresponds to sq-1,2,3,4,5,6,7,8 with z=44 and ${\displaystyle p_{c}=1-(1-\phi _{c})^{1/9}=0.095765(5)\ldots }$. The value of z for a k x k square is (2k+1)²-5.

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

Lattice	${\displaystyle {\overline {z}}}$	${\displaystyle \alpha }$	${\displaystyle d}$	Site percolation threshold	Bond percolation threshold
square		0.2	1.1	0.8025(2)[57]
		0.2	1.2	0.6667(5)[57]
		0.1	1.1	0.6619(1)[57]

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 ${\displaystyle z=k^{2}+10k-2}$ for ${\displaystyle 1\times k}$ sticks.

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

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

In AB percolation, a ${\displaystyle p_{\mathrm {site} }}$ is the proportion of A sites among B sites, and bonds are drawn between sites of opposite species.^[61] It is also called antipercolation.

In colored percolation, occupied sites are assigned one of ${\displaystyle n}$ colors with equal probability, and connection is made along bonds between neighbors of different colors.^[62]

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold
triangular AB	6	6	0.2145,[61] 0.21524(34),[63] 0.21564(3)[64]
AB on square-covering lattice	6	6	${\displaystyle 1-{\sqrt {1-p_{c}(site,sq)}}=0.361835}$[65]
square three-color	4	4	0.80745(5)[62]
square four-color	4	4	0.73415(4)[62]
square five-color	4	4	0.69864(7)[62]
square six-color	4	4	0.67751(5)[62]
triangular two-color	6	6	0.72890(4)[62]
triangular three-color	6	6	0.63005(4)[62]
triangular four-color	6	6	0.59092(3)[62]
triangular five-color	6	6	0.56991(5)[62]
triangular six-color	6	6	0.55679(5)[62]

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

Square lattice:

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
square	4	4	0.615185(15)[66]	0.95
			0.667280(15)[66]	0.85
			0.732100(15)[66]	0.75
			0.75	0.726195(15)[66]
			0.815560(15)[66]	0.65
			0.85	0.615810(30)[66]
			0.95	0.533620(15)[66]

Honeycomb (hexagonal) lattice:

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
honeycomb	3	3	0.7275(5)[67]	0.95
			0. 0.7610(5)[67]	0.90
			0.7986(5)[67]	0.85
			0.80	0.8481(5)[67]
			0.8401(5)[67]	0.80
			0.85	0.7890(5)[67]
			0.90	0.7377(5)[67]
			0.95	0.6926(5)[67]

Kagome lattice:

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
kagome	4	4	0.6711(4),[67] 0.67097(3)[68]	0.95
			0.6914(5),[67] 0.69210(2)[68]	0.90
			0.7162(5),[67] 0.71626(3)[68]	0.85
			0.7428(5),[67] 0.74339(3)[68]	0.80
			0.75	0.7894(9)[67]
			0.7757(8),[67] 0.77556(3)[68]	0.75
			0.80	0.7152(7)[67]
			0.81206(3)[68]	0.70
			0.85	0.6556(6)[67]
			0.85519(3)[68]	0.65
			0.90	0.6046(5)[67]
			0.90546(3)[68]	0.60
			0.95	0.5615(4)[67]
			0.96604(4)[68]	0.55
			0.9854(3)[68]	0.53

* For values on different lattices, see "An investigation of site-bond percolation on many lattices".^[67]

Approximate formula for site-bond percolation on a honeycomb lattice

Lattice	z	${\displaystyle {\overline {z}}}$	Threshold	Notes
(63) honeycomb	3	3	${\displaystyle p_{b}p_{s}[1-({\sqrt {p_{bc}}}/(3-p_{bc}))({\sqrt {p_{b}}}-{\sqrt {p_{bc}}})]=p_{bc}}$, When equal: ps = pb = 0.82199	approximate formula, ps = site prob., pb = bond prob., pbc = 1 − 2 sin (π/18),[19] exact at ps=1, pb=pbc.

Laves lattices are the duals to the Archimedean lattices. Drawings from.^[6] See also Uniform tilings.

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
Cairo pentagonal D(32,4,3,4)=(2⁄3)(53)+(1⁄3)(54)	3,4	3 1⁄3	0.6501834(2),[11] 0.650184(5)[6]	0.585863... = 1 − pcbond(32,4,3,4)
Pentagonal D(33,42)=(1⁄3)(54)+(2⁄3)(53)	3,4	3 1⁄3	0.6470471(2),[11] 0.647084(5),[6] 0.6471(6)[39]	0.580358... = 1 − pcbond(33,42), 0.5800(6)[39]
D(34,6)=(1⁄5)(46)+(4⁄5)(43)	3,6	3 3⁄5	0.639447[6]	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),[69] 0.585040(5)[6]	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)[6]	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,[70] 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)[71] D(3, 122)=(2⁄3)(33)+(1⁄3)(312)	3,12	6	1⁄2	0.259579... = 1 − pcbond(3, 122)

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]

#	Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
41	(1⁄2)(3,4,3,12) + (1⁄2)(3, 122)	4,3	3.5	0.7680(2)[72]	0.67493252(36)[citation needed]
42	(1⁄3)(3,4,6,4) + (2⁄3)(4,6,12)	4,3	31⁄3	0.7157(2)[72]	0.64536587(40)[citation needed]
36	(1⁄7)(36) + (6⁄7)(32,4,12)	6,4	4 2⁄7	0.6808(2)[72]	0.55778329(40)[citation needed]
15	(2⁄3)(32,62) + (1⁄3)(3,6,3,6)	4,4	4	0.6499(2)[72]	0.53632487(40)[citation needed]
34	(1⁄7)(36) + (6⁄7)(32,62)	6,4	4 2⁄7	0.6329(2)[72]	0.51707873(70)[citation needed]
16	(4⁄5)(3,42,6) + (1⁄5)(3,6,3,6)	4,4	4	0.6286(2)[72]	0.51891529(35)[citation needed]
17	(4⁄5)(3,42,6) + (1⁄5)(3,6,3,6)*	4,4	4	0.6279(2)[72]	0.51769462(35)[citation needed]
35	(2⁄3)(3,42,6) + (1⁄3)(3,4,6,4)	4,4	4	0.6221(2)[72]	0.51973831(40)[citation needed]
11	(1⁄2)(34,6) + (1⁄2)(32,62)	5,4	4.5	0.6171(2)[72]	0.48921280(37)[citation needed]
37	(1⁄2)(33,42) + (1⁄2)(3,4,6,4)	5,4	4.5	0.5885(2)[72]	0.47229486(38)[citation needed]
30	(1⁄2)(32,4,3,4) + (1⁄2)(3,4,6,4)	5,4	4.5	0.5883(2)[72]	0.46573078(72)[citation needed]
23	(1⁄2)(33,42) + (1⁄2)(44)	5,4	4.5	0.5720(2)[72]	0.45844622(40)[citation needed]
22	(2⁄3)(33,42) + (1⁄3)(44)	5,4	4 2⁄3	0.5648(2)[72]	0.44528611(40)[citation needed]
12	(1⁄4)(36) + (3⁄4)(34,6)	6,5	5 1⁄4	0.5607(2)[72]	0.41109890(37)[citation needed]
33	(1⁄2)(33,42) + (1⁄2)(32,4,3,4)	5,5	5	0.5505(2)[72]	0.41628021(35)[citation needed]
32	(1⁄3)(33,42) + (2⁄3)(32,4,3,4)	5,5	5	0.5504(2)[72]	0.41549285(36)[citation needed]
31	(1⁄7)(36) + (6⁄7)(32,4,3,4)	6,5	5 1⁄7	0.5440(2)[72]	0.40379585(40)[citation needed]
13	(1⁄2)(36) + (1⁄2)(34,6)	6,5	5.5	0.5407(2)[72]	0.38914898(35)[citation needed]
21	(1⁄3)(36) + (2⁄3)(33,42)	6,5	5 1⁄3	0.5342(2)[72]	0.39491996(40)[citation needed]
20	(1⁄2)(36) + (1⁄2)(33,42)	6,5	5.5	0.5258(2)[72]	0.38285085(38)[citation needed]

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^[73] 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.

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

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

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
martini (3⁄4)(3,92)+(1⁄4)(93)	3	3	0.764826..., 1 + p4 − 3p3 = 0[74]	0.707107... = 1/√2[75]
bow-tie (c)	3,4	3 1⁄7		0.672929..., 1 − 2p3 − 2p4 − 2p5 − 7p6 + 18p7 + 11p8 − 35p9 + 21p10 − 4p11 = 0[76]
bow-tie (d)	3,4	3 1⁄3		0.625457..., 1 − 2p2 − 3p3 + 4p4 − p5 = 0[76]
martini-A (2⁄3)(3,72)+(1⁄3)(3,73)	3,4	3 1⁄3	1/√2[76]	0.625457..., 1 − 2p2 − 3p3 + 4p4 − p5 = 0[76]
bow-tie dual (e)	3,4	3 2⁄3		0.595482..., 1-pcbond (bow-tie (a))[76]
bow-tie (b)	3,4,6	3 2⁄3		0.533213..., 1 − p − 2p3 -4p4-4p5+156+ 13p7-36p8+19p9+ p10 + p11=0[76]
martini covering/medial (1⁄2)(33,9) + (1⁄2)(3,9,3,9)	4	4	0.707107... = 1/√2[75]	0.57086651(33)[citation needed]
martini-B (1⁄2)(3,5,3,52) + (1⁄2)(3,52)	3, 5	4	0.618034... = 2/(1 + √5), 1- p2 − p = 0[74][76]	1⁄2[75][76]
bow-tie dual (f)	3,4,8	4 2⁄5		0.466787..., 1 − pcbond (bow-tie (b))[76]
bow-tie (a) (1⁄2)(32,4,32,4) + (1⁄2)(3,4,3)	4,6	5	0.5472(2),[39] 0.5479148(7)[77]	0.404518..., 1 − p − 6p2 + 6p3 − p5 = 0[76][78]
bow-tie dual (h)	3,6,8	5		0.374543..., 1 − pcbond(bow-tie (d))[76]
bow-tie dual (g)	3,6,10	5 1⁄2	0.547... = pcsite(bow-tie(a))	0.327071..., 1 − pcbond(bow-tie (c))[76]
martini dual (1⁄2)(33) + (1⁄2)(39)	3,9	6	1⁄2	0.292893... = 1 − 1/√2[75]

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
(4, 6, 12) covering/medial	4	4	pcbond(4, 6, 12) = 0.693731...	0.5593140(2),[11] 0.559315(1)[citation needed]
(4, 82) covering/medial, square kagome	4	4	pcbond(4,82) = 0.676803...	0.544798017(4),[11] 0.54479793(34)[citation needed]
(34, 6) medial	4	4		0.5247495(5)[11]
(3,4,6,4) medial	4	4		0.51276[11]
(32, 4, 3, 4) medial	4	4		0.512682929(8)[11]
(33, 42) medial	4	4		0.5125245984(9)[11]
square covering (non-planar)	6	6	1⁄2	0.3371(1)[59]
square matching lattice (non-planar)	8	8	1 − pcsite(square) = 0.407253...	0.25036834(6)[18]

(4, 6, 12) covering/medial lattice

(4, 8²) covering/medial lattice

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

(3,4,6,4) covering/medial lattice, equivalent to the 2-uniform lattice #30, but with facing triangles made into a diamond. This pattern appears in Iranian tilework.^[79] such as Western tomb tower, Kharraqan.^[80]

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

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
K(2,2)	4	4	0.51253(14)[81]	0.44778(15)[81]
K(3,3)	6	6	0.43760(15)[81]	0.35502(15)[81]
K(4,4)	8	8	0.38675(7)[81]	0.29427(12)[81]
K(5,5)	10	10	0.35115(13)[81]	0.25159(13)[81]
K(6,6)	12	12	0.32232(13)[81]	0.21942(11)[81]
K(7,7)	14	14	0.30052(14)[81]	0.19475(9)[81]
K(8,8)	16	16	0.28103(11)[81]	0.17496(10)[81]

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

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
checkerboard – 2 × 2 subnet	4,3			0.596303(1)[83]
checkerboard – 4 × 4 subnet	4,3			0.633685(9)[83]
checkerboard – 8 × 8 subnet	4,3			0.642318(5)[83]
checkerboard – 16 × 16 subnet	4,3			0.64237(1)[83]
checkerboard – 32 × 32 subnet	4,3			0.64219(2)[83]
checkerboard – ${\displaystyle \infty }$ subnet	4,3			0.642216(10)[83]
kagome – 2 × 2 subnet = (3, 122) covering/medial	4		pcbond (3, 122) = 0.74042077...	0.600861966960(2),[11] 0.6008624(10),[19] 0.60086193(3)[9]
kagome – 3 × 3 subnet	4			0.6193296(10),[19] 0.61933176(5),[9] 0.61933044(32)[citation needed]
kagome – 4 × 4 subnet	4			0.625365(3),[19] 0.62536424(7)[9]
kagome – ${\displaystyle \infty }$ subnet	4			0.628961(2)[19]
kagome – (1 × 1):(2 × 2) subnet = martini covering/medial	4		pcbond(martini) = 1/√2 = 0.707107...	0.57086648(36)[citation needed]
kagome – (1 × 1):(3 × 3) subnet	4,3		0.728355596425196...[9]	0.58609776(37)[citation needed]
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)[citation needed]
triangular – 2 × 2 subnet	6,4			0.471628788[83]
triangular – 3 × 3 subnet	6,4			0.509077793[83]
triangular – 4 × 4 subnet	6,4			0.524364822[83]
triangular – 5 × 5 subnet	6,4			0.5315976(10)[83]
triangular – ${\displaystyle \infty }$ subnet	6,4			0.53993(1)[83]

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

system	z	Site threshold
dimers on a honeycomb lattice	3	0.69,[84] 0.6653 [85]
dimers on a triangular lattice	6	0.4872(8),[84] 0.4873,[85]
aligned linear dimers on a triangular lattice	6	0.5157(2)[86]
aligned linear 4-mers on a triangular lattice	6	0.5220(2)[86]
aligned linear 8-mers on a triangular lattice	6	0.5281(5)[86]
aligned linear 12-mers on a triangular lattice	6	0.5298(8)[86]
linear 16-mers on a triangular lattice	6	aligned 0.5328(7)[86]
linear 32-mers on a triangular lattice	6	aligned 0.5407(6)[86]
linear 64-mers on a triangular lattice	6	aligned 0.5455(4)[86]
aligned linear 80-mers on a triangular lattice	6	0.5500(6)[86]
aligned linear k ${\displaystyle \longrightarrow \infty }$ on a triangular lattice	6	0.582(9)[86]
dimers and 5% impurities, triangular lattice	6	0.4832(7)[87]
parallel dimers on a square lattice	4	0.5863[88]
dimers on a square lattice	4	0.5617,[88] 0.5618(1),[89] 0.562,[90] 0.5713[85]
linear 3-mers on a square lattice	4	0.528[90]
3-site 120° angle, 5% impurities, triangular lattice	6	0.4574(9)[87]
3-site triangles, 5% impurities, triangular lattice	6	0.5222(9)[87]
linear trimers and 5% impurities, triangular lattice	6	0.4603(8)[87]
linear 4-mers on a square lattice	4	0.504[90]
linear 5-mers on a square lattice	4	0.490[90]
linear 6-mers on a square lattice	4	0.479[90]
linear 8-mers on a square lattice	4	0.474,[90] 0.4697(1)[89]
linear 10-mers on a square lattice	4	0.469[90]
linear 16-mers on a square lattice	4	0.4639(1)[89]
linear 32-mers on a square lattice	4	0.4747(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]

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

System is composed of ordinary (non-avoiding) random walks of length l on the square lattice.^[93]

Lattice	z	Site percolation threshold	Bond percolation threshold
bow-tie with p = 1⁄2 on one non-diagonal bond	3		0.3819654(5),[96] ${\displaystyle (3-{\sqrt {5}})/2}$[59]

System	Φc	ηc	nc
Disks of radius r	0.67634831(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]		${\displaystyle \rho _{c}r^{2}}$ = 1.108010(7)[115]
Ellipses, ε = 1.5	0.0043[103]	0.00431	2.059081(7)[112]
Ellipses, ε = 5⁄3	0.65[116]	1.05[116]	2.28[116]
Ellipses, ε = 2	0.6287945(12),[112] 0.63[116]	0.991000(3),[112] 0.99[116]	2.523560(8),[112] 2.5[116]
Ellipses, ε = 3	0.56[116]	0.82[116]	3.157339(8),[112] 3.14[116]
Ellipses, ε = 4	0.5[116]	0.69[116]	3.569706(8),[112] 3.5[116]
Ellipses, ε = 5	0.455,[103] 0.455,[105] 0.46[116]	0.607[103]	3.861262(12),[112] 3.86[103]
Ellipses, ε = 6			4.079365(17)[112]
Ellipses, ε = 7			4.249132(16)[112]
Ellipses, ε = 8			4.385302(15)[112]
Ellipses, ε = 9			4.497000(8)[112]
Ellipses, ε = 10	0.301,[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 = 15			4.894752(30)[112]
Ellipses, ε = 20	0.178,[103] 0.17[116]	0.196[103]	5.062313(39),[112] 4.99[103]
Ellipses, ε = 50	0.081[103]	0.084[103]	5.393863(28),[112] 5.38[103]
Ellipses, ε = 100	0.0417[103]	0.0426[103]	5.513464(40),[112] 5.42[103]
Ellipses, ε = 200	0.021[116]	0.0212[116]	5.40[116]
Ellipses, ε = 1000	0.0043[103]	0.00431	5.624756(22),[112] 5.5
Superellipses, ε = 1, m = 1.5	0.671[105]
Superellipses, ε = 2.5, m = 1.5	0.599[105]
Superellipses, ε = 5, m = 1.5	0.469[105]
Superellipses, ε = 10, m = 1.5	0.322[105]
disco-rectangles, ε = 1.5			1.894 [111]
disco-rectangles, ε = 2			2.245 [111]
Aligned squares of side ${\displaystyle \ell }$	0.66675(2),[54] 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 squares	0.62554075(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 ${\displaystyle \pi /4}$	0.6255(1)[118]	0.98216(15)[118]
Rectangles, ε = 1.1	0.624870(7)	0.980484(19)	1.078532(21)[119]
Rectangles, ε = 2	0.590635(5)	0.893147(13)	1.786294(26)[119]
Rectangles, ε = 3	0.5405983(34)	0.777830(7)	2.333491(22)[119]
Rectangles, ε = 4	0.4948145(38)	0.682830(8)	2.731318(30)[119]
Rectangles, ε = 5	0.4551398(31), 0.451[105]	0.607226(6)	3.036130(28)[119]
Rectangles, ε = 10	0.3233507(25), 0.319[105]	0.3906022(37)	3.906022(37)[119]
Rectangles, ε = 20	0.2048518(22)	0.2292268(27)	4.584535(54)[119]
Rectangles, ε = 50	0.09785513(36)	0.1029802(4)	5.149008(20)[119]
Rectangles, ε = 100	0.0523676(6)	0.0537886(6)	5.378856(60)[119]
Rectangles, ε = 200	0.02714526(34)	0.02752050(35)	5.504099(69)[119]
Rectangles, ε = 1000	0.00559424(6)	0.00560995(6)	5.609947(60)[119]
Sticks (needles) of length ${\displaystyle \ell }$			5.63726(2),[120] 5.6372858(6),[97] 5.637263(11),[119] 5.63724(18)[121]
sticks with log-normal length dist. STD=0.5			4.756(3)[121]
sticks with correlated angle dist. s=0.5			6.6076(4)[121]
Power-law disks, x = 2.05	0.993(1)[122]	4.90(1)	0.0380(6)
Power-law disks, x = 2.25	0.8591(5)[122]	1.959(5)	0.06930(12)
Power-law disks, x = 2.5	0.7836(4)[122]	1.5307(17)	0.09745(11)
Power-law disks, x = 4	0.69543(6)[122]	1.18853(19)	0.18916(3)
Power-law disks, x = 5	0.68643(13)[122]	1.1597(3)	0.22149(8)
Power-law disks, x = 6	0.68241(8)[122]	1.1470(1)	0.24340(5)
Power-law disks, x = 7	0.6803(8)[122]	1.140(6)	0.25933(16)
Power-law disks, x = 8	0.67917(9)[122]	1.1368(5)	0.27140(7)
Power-law disks, x = 9	0.67856(12)[122]	1.1349(4)	0.28098(9)
Voids around disks of radius r	1 − Φc(disk) = 0.32355169(2),[97] 0.318(2),[123] 0.3261(6)[124]

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

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

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

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

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

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

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

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

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

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

For ellipses, ${\displaystyle n_{c}=(4\epsilon /\pi )\eta _{c}}$

For void percolation, ${\displaystyle \phi _{c}=e^{-\eta _{c}}}$ 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 ${\displaystyle |x/a|^{2m}+|y/b|^{2m}=1}$. 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 ^[125]

For binary dispersions of disks, see ^{[98][126][115]}

Lattice	z	${\displaystyle {\overline {z}}}$	Site percolation threshold	Bond percolation threshold
Relative neighborhood graph		2.5576	0.796(2)[127]	0.771(2)[127]
Voronoi tessellation	3		0.71410(2),[129] 0.7151*[72]	0.68,[130] 0.6670(1),[131] 0.6680(5),[132] 0.666931(5)[129]
Voronoi covering/medial	4		0.666931(2)[129][131]	0.53618(2)[129]
Randomized kagome/square-octagon, fraction r=1⁄2	4		0.6599[16]
Penrose rhomb dual	4		0.6381(3)[69]	0.5233(2)[69]
Gabriel graph		4	0.6348(8),[133] 0.62[134]	0.5167(6),[133] 0.52[134]
Random-line tessellation, dual		4	0.586(2)[135]
Penrose rhomb		4	0.5837(3),[69] 0.0.5610(6) (weighted bonds)[136] 0.58391(1)[137]	0.483(5),[138] 0.4770(2)[69]
Octagonal lattice, "chemical" links (Ammann–Beenker tiling)		4	0.585[139]	0.48[139]
Octagonal lattice, "ferromagnetic" links		5.17	0.543[139]	0.40[139]
Dodecagonal lattice, "chemical" links		3.63	0.628[139]	0.54[139]
Dodecagonal lattice, "ferromagnetic" links		4.27	0.617[139]	0.495[139]
Delaunay triangulation		6	1⁄2[140]	0.3333(1)[131] 0.3326(5),[132] 0.333069(2)[129]
Uniform Infinite Planar Triangulation[141]		6	1⁄2	(2√3 – 1)/11 ≈ 0.2240[128][142]