Percolation threshold

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Threshold of percolation theory models

title: "Percolation threshold" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["percolation-theory", "critical-phenomena", "random-graphs"] description: "Threshold of percolation theory models" topic_path: "general/percolation-theory" source: "https://en.wikipedia.org/wiki/Percolation_threshold" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Threshold of percolation theory models ::

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.

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)= 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. | last1 = Kasteleyn | first1 = P. W. | first2 = C. M. | last2 = Fortuin | title = Phase transitions in lattice systems with random local properties | journal = Journal of the Physical Society of Japan Supplement | volume = 26 | year = 1969 | pages = 11–14| bibcode = 1969JPSJS..26...11K 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 , and self-dual lattices (square, martini-B) have bond thresholds of .

The notation such as (4,82) comes from Grünbaum and Shephard, 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 , the critical point gets scaled by (1-C)^{-1} such that:

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.

Percolation in 2D

Thresholds on Archimedean lattices

::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/9e/Archimedean-Lattice.png" caption="This is a picture"] ::

::data[format=table]

Lattice	z	\overline z	Site percolation threshold	Bond percolation threshold
3-12 or super-kagome, (3, 122 )	3	3	1	2}}
cross, truncated trihexagonal (4, 6, 12)	3	3	0.746, 0.750, 0.747806(4), 0.7478008(2)	0.6937314(1), 0.69373383(72), 0.693733124922(2)
square octagon, bathroom tile, 4-8, truncated square	3	3	-	0.729, 0.729724(3), 0.7297232(5)
honeycomb (63)	3	3	0.6962(6), 0.697040230(5), 0.6970402(1), 0.6970413(10), 0.697043(3),	0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=0
kagome (3, 6, 3, 6)	4	4	0.652703645... = 1 − 2 sin(/18)	0.5244053(3), 0.52440516(10), 0.52440499(2), 0.524404978(5), 0.52440572..., 0.52440500(1), 0.524404999173(3), 0.524404999167439(4) 0.52440499916744820(1)
rhombitrihexagonal]] (3, 4, 6, 4)	4	4	0.620, 0.621819(3), 0.62181207(7)	0.52483258(53), 0.5248311(1), 0.524831461573(1)
square (44)	4	4	last=Mertens	first=Stephan
snub hexagonal, maple leaf (34,6)	5	5	0.579 0.579498(3)	0.43430621(50), 0.43432764(3), 0.4343283172240(6),
snub square, puzzle (32, 4, 3, 4 )	5	5	0.550, 0.550806(3)	0.41413743(46), 0.4141378476(7), 0.4141378565917(1),
frieze, elongated triangular(33, 42)	5	5	0.549, 0.550213(3), 0.5502(8)	0.4196(6), 0.41964191(43), 0.41964044(1), 0.41964035886369(2)
triangular (36)	6	6		0.347296355... = 2 sin (/18), 1 + p3 − 3p = 0
::

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), etc. Equivalent to square-2N+3N+4N, sq(1,2,3). tri = triangular, hc = honeycomb. ::data[format=table]

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

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.

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). 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 x k square is (2k+1)2-5.

2D distorted lattices

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

::data[format=table]

Lattice	\alpha	d	Site percolation threshold
square	0.2	1.1	0.8025(2)
	0.2	1.2	0.6667(5)
	0.1	1.1	0.6619(1)
::

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.

::data[format=table]

System	k	z	Site coverage φc	Site percolation threshold pc
1 x 2 dimer, square lattice	2	22	0.54691	last = Jasna
1 x 2 aligned dimer, square lattice	2	14(?)	0.5715(18)	0.3454(13)
1 x 3 trimer, square lattice	3	37	0.49898	0.10880(2)
1 x 4 stick, square lattice	4	54	0.45761	0.07362(2)
1 x 5 stick, square lattice	5	73	0.42241	0.05341(1)
1 x 6 stick, square lattice	6	94	0.39219	0.04063(2)
::

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

::data[format=table]

Lattice	z	Site percolation threshold	Bond percolation threshold
(3, 122 )	3
(4, 6, 12)	3
(4, 82)	3		0.676835..., 4p3 + 3p4 − 6 p5 − 2 p6 = 1
honeycomb (63)	3
kagome (3, 6, 3, 6)	4		0.524430..., 3p2 + 6p3 − 12 p4+ 6 p5 − p6 = 1
(3, 4, 6, 4)	4
square (44)	4		(exact)
(34,6 )	5		date=December 2018}}
snub square, puzzle (32, 4, 3, 4 )	5
(33, 42)	5
triangular (36)	6	(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. 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.

::data[format=table]

Lattice	z	\overline z	Site percolation threshold
triangular AB	6	6	last=Nakanishi
AB on square-covering lattice	6	6	last1=Wu
square three-color	4	4	last1=Kundu
square four-color	4	4	0.73415(4)
square five-color	4	4	0.69864(7)
square six-color	4	4	0.67751(5)
triangular two-color	6	6	0.72890(4)
triangular three-color	6	6	0.63005(4)
triangular four-color	6	6	0.59092(3)
triangular five-color	6	6	0.56991(5)
triangular six-color	6	6	0.55679(5)
::

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: ::data[format=table]

Lattice	z	\overline z	Site percolation threshold	Bond percolation threshold
square	4	4	0.615185(15)	0.95
			0.667280(15)	0.85
			0.732100(15)	0.75
			0.75	0.726195(15)
			0.815560(15)	0.65
			0.85	0.615810(30)
			0.95	0.533620(15)
::

Honeycomb (hexagonal) lattice: ::data[format=table]

Lattice	z	\overline z	Site percolation threshold	Bond percolation threshold
honeycomb	3	3	0.7275(5)	0.95
			0. 0.7610(5)	0.90
			0.7986(5)	0.85
			0.80	0.8481(5)
			0.8401(5)	0.80
			0.85	0.7890(5)
			0.90	0.7377(5)
			0.95	0.6926(5)
::

Kagome lattice: ::data[format=table]

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

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

Approximate formula for site-bond percolation on a honeycomb lattice

::data[format=table]

Lattice	z	\overline z	Threshold	Notes
(63) honeycomb	3	3	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 (/18), exact at ps=1, pb=pbc.
::

Archimedean duals (Laves lattices)

::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/15/Dual_Archimedean.png" caption="Example image caption"] ::

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

::data[format=table]

Lattice	z	\overline z	Site percolation threshold	Bond percolation threshold
Cairo pentagonal	3,4	3	0.6501834(2), 0.650184(5)	0.585863... = 1 − pcbond(32,4,3,4)
1	3}})(54)+()(53)	3,4	3	0.6470471(2), 0.647084(5), 0.6471(6)
1	5}})(46)+()(43)	3,6	3	0.639447
dice, rhombille tiling	3,6	4	0.5851(4), 0.585040(5)	0.475595... = 1 − pcbond(3,6,3,6 )
ruby dual	3,4,6	4	0.582410(5)	0.475167... = 1 − pcbond(3,4,6,4 )
union jack, tetrakis square tiling	4,8	6		0.323197... = 1 − pcbond(4,82 )
bisected hexagon, cross dual	4,6,12	6		0.306266... = 1 − pcbond(4,6,12)
asanoha (hemp leaf)	3,12	6		0.259579... = 1 − pcbond(3, 122)
::

2-uniform lattices

Top 3 lattices: #13 #12 #36 Bottom 3 lattices: #34 #37 #11 ::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/9c/2uni4m1.png" caption="20 2 uniform lattices"] ::

Top 2 lattices: #35 #30 Bottom 2 lattices: #41 #42 ::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/13/2uni4m2.png" caption="20 2 uniform lattices"] ::

Top 4 lattices: #22 #23 #21 #20 Bottom 3 lattices: #16 #17 #15 ::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/12/2uni4m3.png" caption="20 2 uniform lattices"] ::

Top 2 lattices: #31 #32 Bottom lattice: #33 ::figure[src="https://upload.wikimedia.org/wikipedia/commons/2/20/2uni4m4.png" caption="20 2 uniform lattices"] ::

::data[format=table]

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

Inhomogeneous 2-uniform lattice

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/07/2uniformLattice37.pdf" caption="2-uniform lattice #37"] ::

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 ()(33,42) + ()(3,4,6,4), while the dual lattice has vertex types ()(46)+()(42,52)+()(53)+()(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 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).

::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/10/Martini.png" caption="Example image caption"] ::

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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/bb/Bow-tie.png" caption="Example image caption"] ::

::data[format=table]

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

Thresholds on 2D covering, medial, and matching lattices

::data[format=table]

Lattice	z	\overline z	Site percolation threshold	Bond percolation threshold
(4, 6, 12) covering/medial	4	4	pcbond(4, 6, 12) = 0.693731...	date=June 2019}}
(4, 82) covering/medial, square kagome	4	4	pcbond(4,82) = 0.676803...	date=June 2019}}
(34, 6) medial	4	4		0.5247495(5)
(3,4,6,4) medial	4	4		0.51276
(32, 4, 3, 4) medial	4	4		0.512682929(8)
(33, 42) medial	4	4		0.5125245984(9)
square covering (non-planar)	6	6		0.3371(1)
square matching lattice (non-planar)	8	8	1 − pcsite(square) = 0.407253...	0.25036834(6)
::

|image1=4,6,12covering.svg |caption1=(4, 6, 12) covering/medial lattice |image2=4,82coveringlattice.svg |caption2=(4, 82) covering/medial lattice |image3=312coveringdual.svg |caption3=(3,122) covering/medial lattice (in light grey), equivalent to the kagome (2 × 2) subnet, and in black, the dual of these lattices.

|image1=(3,4,6,4) medial lattice.svg |caption1=(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. such as Western tomb tower, Kharraqan. |image2=(3,4,6,4) medial dual.svg |caption2=(3,4,6,4) medial dual, shown in red, with medial lattice in light gray behind it

Thresholds on 2D chimera non-planar lattices

::data[format=table]

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

Thresholds on subnet lattices

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/6b/Kagomesubnets.png" caption="Example image caption"] ::

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.{{cite journal | last = Okubo | first = S. | author2 = M. Hayashi |author3=S. Kimura |author4=H. Ohta |author5=M. Motokawa |author6=H. Kikuchi |author7=H. Nagasawa | title = Submillimeter wave ESR of triangular-kagome antiferromagnet Cu9X2(cpa)6 (X=Cl, Br) | journal = Physica B | volume = 246–247 | issue = 2 | year = 1998 | doi = 10.1016/S0921-4526(97)00985-X | pages = 553–556|bibcode = 1998PhyB..246..553O }}

::data[format=table]

Lattice	z	Site percolation threshold	Bond percolation threshold
checkerboard – 2 × 2 subnet	4,3		0.596303(1){{cite journal
checkerboard – 4 × 4 subnet	4,3		0.633685(9)
checkerboard – 8 × 8 subnet	4,3		0.642318(5)
checkerboard – 16 × 16 subnet	4,3		0.64237(1)
checkerboard – 32 × 32 subnet	4,3		0.64219(2)
checkerboard – \infty subnet	4,3		0.642216(10)
kagome – 2 × 2 subnet = (3, 122) covering/medial	4	pcbond (3, 122) = 0.74042077...	0.600861966960(2), 0.6008624(10), 0.60086193(3)
kagome – 3 × 3 subnet	4		date=June 2019}}
kagome – 4 × 4 subnet	4		0.625365(3), 0.62536424(7)
kagome – \infty subnet	4		0.628961(2)
kagome – (1 × 1):(2 × 2) subnet = martini covering/medial	4	2}} = 0.707107...	0.57086648(36)
kagome – (1 × 1):(3 × 3) subnet	4,3	0.728355596425196...	0.58609776(37)
kagome – (1 × 1):(4 × 4) subnet		0.738348473943256...
kagome – (1 × 1):(5 × 5) subnet		0.743548682503071...
kagome – (1 × 1):(6 × 6) subnet		0.746418147634282...
kagome – (2 × 2):(3 × 3) subnet			0.61091770(30)
triangular – 2 × 2 subnet	6,4		0.471628788
triangular – 3 × 3 subnet	6,4		0.509077793
triangular – 4 × 4 subnet	6,4		0.524364822
triangular – 5 × 5 subnet	6,4		0.5315976(10)
triangular – \infty subnet	6,4		0.53993(1)
::

Thresholds of random sequentially adsorbed objects

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

::data[format=table]

system	z	Site threshold
dimers on a honeycomb lattice	3	0.69, 0.6653 {{cite journal
dimers on a triangular lattice	6	0.4872(8),{{cite journal
aligned linear dimers on a triangular lattice	6	0.5157(2) {{cite journal
aligned linear 4-mers on a triangular lattice	6	0.5220(2)
aligned linear 8-mers on a triangular lattice	6	0.5281(5)
aligned linear 12-mers on a triangular lattice	6	0.5298(8)
linear 16-mers on a triangular lattice	6	aligned 0.5328(7)
linear 32-mers on a triangular lattice	6	aligned 0.5407(6)
linear 64-mers on a triangular lattice	6	aligned 0.5455(4)
aligned linear 80-mers on a triangular lattice	6	0.5500(6)
aligned linear k \longrightarrow \infty on a triangular lattice	6	0.582(9)
dimers and 5% impurities, triangular lattice	6	0.4832(7){{cite journal
parallel dimers on a square lattice	4	0.5863
dimers on a square lattice	4	0.5617,{{cite journal
linear 3-mers on a square lattice	4	0.528
3-site 120° angle, 5% impurities, triangular lattice	6	0.4574(9)
3-site triangles, 5% impurities, triangular lattice	6	0.5222(9)
linear trimers and 5% impurities, triangular lattice	6	0.4603(8)
linear 4-mers on a square lattice	4	0.504
linear 5-mers on a square lattice	4	0.490
linear 6-mers on a square lattice	4	0.479
linear 8-mers on a square lattice	4	0.474, 0.4697(1)
linear 10-mers on a square lattice	4	0.469
linear 16-mers on a square lattice	4	0.4639(1)
linear 32-mers on a square lattice	4	0.4747(2)
::

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.{{cite journal | last =Kondrat | first = Grzegorz |author2=Andrzej Pękalski | title = Percolation and jamming in random sequential adsorption of linear segments on a square lattice | journal = Physical Review E | volume = 63 | issue = 5 | year = 2001 | article-number = 051108 | doi = 10.1103/PhysRevE.63.051108| pmid = 11414888 | arxiv =cond-mat/0102031| bibcode =2001PhRvE..63e1108K| s2cid = 44490067

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.

::data[format=table]

system	z	Bond threshold
Parallel covering, square lattice	6	0.381966...{{cite journal
Shifted covering, square lattice	6	0.347296...
Staggered covering, square lattice	6	0.376825(2)
Random covering, square lattice	6	0.367713(2)
Parallel covering, triangular lattice	10	0.237418...
Staggered covering, triangular lattice	10	0.237497(2)
Random covering, triangular lattice	10	0.235340(1)
::

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.{{cite journal | last = Zia | first = R. K. P. |author2=W. Yong |author3=B. Schmittmann | author3-link = Beate Schmittmann | title = Percolation of a collection of finite random walks: a model for gas permeation through thin polymeric membranes | journal = Journal of Mathematical Chemistry | volume = 45 | year = 2009 | pages = 58–64 | doi = 10.1007/s10910-008-9367-6 | s2cid = 94092783 ::data[format=table]

l (polymer length)	z	Bond percolation
1	4	0.5(exact){{cite journal
2	4	0.47697(4)
4	4	0.44892(6)
8	4	0.41880(4)
::

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

::data[format=table]

k	z	Site thresholds	Bond thresholds
1	4	0.593(2){{cite journal	last = Cornette
2	4	0.564(2)	0.4859(2)
3	4	0.552(2)	0.4732(2)
4	4	0.542(2)	0.4630(2)
5	4	0.531(2)	0.4565(2)
6	4	0.522(2)	0.4497(2)
7	4	0.511(2)	0.4423(2)
8	4	0.502(2)	0.4348(2)
9	4	0.493(2)	0.4291(2)
10	4	0.488(2)	0.4232(2)
11	4	0.482(2)	0.4159(2)
12	4	0.476(2)	0.4114(2)
13	4	0.471(2)	0.4061(2)
14	4	0.467(2)	0.4011(2)
15	4	0.4011(2)	0.3979(2)
::

Thresholds on 2D inhomogeneous lattices

::data[format=table]

Lattice	z	Site percolation threshold	Bond percolation threshold
1	2}} on one non-diagonal bond	3
::

Thresholds for 2D continuum models

::data[format=table]

System	Φc	ηc	nc
Disks of radius r	0.67634831(2), 0.6763475(6),{{cite journal	last = Quintanilla	first = John A.
Disks of uniform radius (0,r)	0.686610(7), 0.6860(12), 0.680		\rho_c r^2 = 1.108010(7)
Ellipses, ε = 1.5	0.0043	0.00431	2.059081(7)
5	3}}	0.65	1.05
Ellipses, ε = 2	0.6287945(12), 0.63{{cite journal	last = Xia	first = W.
Ellipses, ε = 3	0.56	0.82	3.157339(8), 3.14
Ellipses, ε = 4	0.5	0.69	3.569706(8), 3.5
Ellipses, ε = 5	0.455,{{cite journal	last = Yi	first = Y.-B.
Ellipses, ε = 6			4.079365(17)
Ellipses, ε = 7			4.249132(16)
Ellipses, ε = 8			4.385302(15)
Ellipses, ε = 9			4.497000(8)
Ellipses, ε = 10	0.301, 0.303, 0.30	0.358 0.36	4.590416(23) 4.56, 4.5
Ellipses, ε = 15			4.894752(30)
Ellipses, ε = 20	0.178, 0.17	0.196	5.062313(39), 4.99
Ellipses, ε = 50	0.081	0.084	5.393863(28), 5.38
Ellipses, ε = 100	0.0417	0.0426	5.513464(40), 5.42
Ellipses, ε = 200	0.021	0.0212	5.40
	0.0043	0.00431	5.624756(22), 5.5
Superellipses, ε = 1, m = 1.5	0.671
Superellipses, ε = 2.5, m = 1.5	0.599
Superellipses, ε = 5, m = 1.5	0.469
Superellipses, ε = 10, m = 1.5	0.322
disco-rectangles, ε = 1.5			1.894
disco-rectangles, ε = 2			2.245
Aligned squares of side \ell	0.66675(2), 0.66674349(3),{{cite journal	last = Mertens	first = Stephan
Randomly oriented squares	0.62554075(4), 0.6254(2) 0.625,	0.9822723(1), 0.9819(6) 0.982278(14){{cite journal	last = Li
Randomly oriented squares within angle \pi/4	0.6255(1)	0.98216(15)
Rectangles, ε = 1.1	0.624870(7)	0.980484(19)	1.078532(21)
Rectangles, ε = 2	0.590635(5)	0.893147(13)	1.786294(26)
Rectangles, ε = 3	0.5405983(34)	0.777830(7)	2.333491(22)
Rectangles, ε = 4	0.4948145(38)	0.682830(8)	2.731318(30)
Rectangles, ε = 5	0.4551398(31), 0.451	0.607226(6)	3.036130(28)
Rectangles, ε = 10	0.3233507(25), 0.319	0.3906022(37)	3.906022(37)
Rectangles, ε = 20	0.2048518(22)	0.2292268(27)	4.584535(54)
Rectangles, ε = 50	0.09785513(36)	0.1029802(4)	5.149008(20)
Rectangles, ε = 100	0.0523676(6)	0.0537886(6)	5.378856(60)
Rectangles, ε = 200	0.02714526(34)	0.02752050(35)	5.504099(69)
Rectangles, ε = 1000	0.00559424(6)	0.00560995(6)	5.609947(60)
Sticks (needles) of length \ell			5.63726(2),{{cite journal
sticks with log-normal length dist. STD=0.5			4.756(3)
sticks with correlated angle dist. s=0.5			6.6076(4)
Power-law disks, x = 2.05	0.993(1)	first = V.	year = 2013
Power-law disks, x = 2.25	0.8591(5)	1.959(5)	0.06930(12)
Power-law disks, x = 2.5	0.7836(4)	1.5307(17)	0.09745(11)
Power-law disks, x = 4	0.69543(6)	1.18853(19)	0.18916(3)
Power-law disks, x = 5	0.68643(13)	1.1597(3)	0.22149(8)
Power-law disks, x = 6	0.68241(8)	1.1470(1)	0.24340(5)
Power-law disks, x = 7	0.6803(8)	1.140(6)	0.25933(16)
Power-law disks, x = 8	0.67917(9)	1.1368(5)	0.27140(7)
Power-law disks, x = 9	0.67856(12)	1.1349(4)	0.28098(9)
Voids around disks of radius r	1 − Φc(disk) = 0.32355169(2), 0.318(2), 0.3261(6){{cite journal	last = Jin	first = Yuliang
::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/bf/2D_continuum_percolation_with_disks.jpg" caption="2D continuum percolation with disks"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/72/2D_continuum_percolation_with_ellipses_of_aspect_ratio_2.jpg" caption="2D continuum percolation with ellipses of aspect ratio 2"] ::

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

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

For disks, Ref. use \phi_c = 1 - e^{-\pi x / 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

For more rectangle values, see

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

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

For binary dispersions of disks, see {{cite journal | last = Meeks | first = Kelsey | author2=J. Tencer | author3=M.L. Pantoya | title = Percolation of binary disk systems: Modeling and theory | journal = Physical Review E | volume = 95 | issue = 1 | year = 2017 | article-number = 012118 | doi = 10.1103/PhysRevE.95.012118 | pmid = 28208494 | bibcode = 2017PhRvE..95a2118M | doi-access = free | last = Quintanilla | first = John A. | title = Measurement of the percolation threshold for fully penetrable disks of different radii | journal = Physical Review E | volume = 63 | issue = 6 | year = 2001 | article-number = 061108 | doi = 10.1103/PhysRevE.63.061108 | pmid = 11415069 | bibcode = 2001PhRvE..63f1108Q

Thresholds on 2D random and quasi-lattices

::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/8d/VoronoiDelaunay.svg" caption="Voronoi diagram (solid lines) and its dual, the Delaunay triangulation (dotted lines), for a [[Poisson distribution]] of points"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a3/Delaunay_triangulation_example.png" caption="Delaunay triangulation"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/69/VoronoiCov12.png" caption="The Voronoi covering or line graph (dotted red lines) and the Voronoi diagram (black lines)"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/4/4e/RNGonDelaunayTriangulation128vertices.jpg" caption="The Relative Neighborhood Graph (black lines) superimposed on the Delaunay triangulation (black plus grey lines)."] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/1b/Gabriel_Graph.png" caption="The Gabriel Graph, a subgraph of the Delaunay triangulation in which the circle surrounding each edge does not enclose any other points of the graph"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/bc/UniformInfinitePlanarTriangulation.png" caption="Uniform Infinite Planar Triangulation, showing bond clusters. From"] ::

::data[format=table]

Lattice	z	\overline z	Site percolation threshold	Bond percolation threshold
Relative neighborhood graph		2.5576	0.796(2){{cite journal	last = Melchert
Voronoi tessellation	3		0.71410(2),{{cite journal	last = Becker
Voronoi covering/medial	4		0.666931(2)	0.53618(2)
1	2}}	4		0.6599
Penrose rhomb dual	4		0.6381(3)	0.5233(2)
Gabriel graph		4	0.6348(8),	first = C.
Random-line tessellation, dual		4	0.586(2){{cite journal	last = Lepage
Penrose rhomb		4	0.5837(3), 0.0.5610(6) (weighted bonds){{cite journal	last = Zhang
Octagonal lattice, "chemical" links (Ammann–Beenker tiling)		4	0.585{{cite journal	last = Babalievski
Octagonal lattice, "ferromagnetic" links		5.17	0.543	0.40
Dodecagonal lattice, "chemical" links		3.63	0.628	0.54
Dodecagonal lattice, "ferromagnetic" links		4.27	0.617	0.495
Delaunay triangulation		6	{{cite journal	last = Bollobás
Uniform Infinite Planar Triangulation{{cite journal	last = Angel	first = Omer	author2 = Schramm, Oded	title = Uniform infinite planar triangulation
::

*Theoretical estimate

Thresholds on 2D correlated systems

Assuming power-law correlations C(r) \sim |r|^{-\alpha} ::data[format=table]

α	Site percolation threshold	Bond percolation threshold
lattice
square	3	0.561406(4){{cite journal
square	2	0.550143(5)
square	0.1	0.508(4)
::

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.

::data[format=table]

Lattice	h	z	\overline z	Site percolation threshold	Bond percolation threshold
simple cubic (open b.c.)	2	5	5	last1=Horton	first1=M. K.
bcc (open b.c.)	2			0.4155
hcp (open b.c.)	2			0.2828
diamond (open b.c.)	2			0.5451
simple cubic (open b.c.)	3			0.4264
bcc (open b.c.)	3			0.3531
bcc (periodic b.c.)	3				0.21113018(38)
hcp (open b.c.)	3			0.2548
diamond (open b.c.)	3			0.5044
simple cubic (open b.c.)	4			0.3997,{{cite journal	last = Sotta
bcc (open b.c.)	4			0.3232
bcc (periodic b.c.)	4				0.20235168(59)
hcp (open b.c.)	4			0.2405
diamond (open b.c.)	4			0.4842
simple cubic (periodic b.c.)	5	6	6		0.278102(5)
simple cubic (open b.c.)	6			0.3708
simple cubic (periodic b.c.)	6	6	6		0.272380(2)
bcc (open b.c.)	6			0.2948
hcp (open b.c.)	6			0.2261
diamond (open b.c.)	6			0.4642
simple cubic (periodic b.c.)	7	6	6	0.3459514(12){{cite journal	last = Gliozzi
simple cubic (open b.c.)	8			0.3557, 0.3565
simple cubic (periodic b.c.)	8	6	6		0.265615(5)
bcc (open b.c.)	8			0.2811
hcp (open b.c.)	8			0.2190
diamond (open b.c.)	8			0.4549
simple cubic (open b.c.)	12			0.3411
bcc (open b.c.)	12			0.2688
hcp (open b.c.)	12			0.2117
diamond (open b.c.)	12			0.4456
simple cubic (open b.c.)	16			0.3219, 0.3339
bcc (open b.c.)	16			0.2622
hcp (open b.c.)	16			0.2086
diamond (open b.c.)	16			0.4415
simple cubic (open b.c.)	32			0.3219,
simple cubic (open b.c.)	64			0.3165,
simple cubic (open b.c.)	128			0.31398,
::

Percolation in 3D

::data[format=table]

Lattice	z	\overline z	filling factor*	filling fraction*	Site percolation threshold	Bond percolation threshold
(10,3)-a oxide (or site-bond)	last = Yoo	first = Ted Y.	author3=Shane P. Stahlheber	author4=Carina E. Kaainoa	author5=Kevin Djepang	author6=Alexander R. Small
(10,3)-b oxide (or site-bond)	23 32	2.4	0.233	last = Wells	first = A. F.	title = Structures Based on the 3-Connected Net 103 – b
silicon dioxide (diamond site-bond)	4,22	2			0.638683(35)
Modified (10,3)-b	32,2	2				0.627
(8,3)-a	3	3			0.577962(33){{cite journal	last = Tran
(10,3)-a gyroid{{cite journal	last = Hyde	first = Stephen T.	author3=Proserpio, Davide M.	title = A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics	journal = Angew. Chem. Int. Ed.	volume = 47
(10,3)-b	3	3			0.565442(40)	0.546694(33)
cubic oxide (cubic site-bond)	6,23	3.5			0.524652(50)
bcc dual		4			0.4560(6)	0.4031(6)
ice Ih	4	4	π / 16 = 0.340087	0.147	0.433(11){{cite journal	last = Frisch
diamond (Ice Ic)	4	4	π / 16 = 0.340087	0.1462332	0.4299(8), 0.4299870(4),{{cite journal	last = Xu
diamond dual		6			0.3904(5)	last = van der Marck
3D kagome (covering graph of the diamond lattice)	6		π / 12 = 0.37024	0.1442	0.3895(2) =pc(site) for diamond dual and pc(bond) for diamond lattice	0.2709(6)
Bow-tie stack dual		5			0.3480(4)	0.2853(4)
honeycomb stack	5	5			0.3701(2)	0.3093(2)
octagonal stack dual	5	5			0.3840(4)	0.3168(4)
pentagonal stack		5			0.3394(4)	0.2793(4)
kagome stack	6	6	0.453450	0.1517	0.3346(4)	0.2563(2)
fcc dual	42,8	5			0.3341(5)	0.2703(3)
simple cubic	6	6	π / 6 = 0.5235988	0.1631574	0.307(10), 0.307, 0.3115(5),{{cite journal	last = Sur
hcp dual	44,82	5			0.3101(5)	0.2573(3)
dice stack	5,8	6	π / 9 = 0.604600	0.1813	0.2998(4)	0.2378(4)
bow-tie stack	7	7			0.2822(6)	0.2092(4)
Stacked triangular / simple hexagonal	8	8			0.26240(5),{{cite journal	last = Schrenk
octagonal (union-jack) stack	6,10	8			0.2524(6)	0.1752(2)
bcc	8	8			0.243(10), 0.243, 0.2459615(10), 0.2460(3),{{cite journal	last = Bradley
simple cubic with 3NN (same as bcc)	8	8			0.2455(1), 0.2457(7){{cite journal	last = Gallyamov
fcc, D3	12	12	π / (3 ) = 0.740480	0.147530	0.195, 0.198(3),{{cite journal	last = Sykes
hcp	12	12	π / (3 ) = 0.740480	0.147545	0.195(5), 0.1992555(10){{cite journal	last = Lorenz
La2−x Srx Cu O4	12	12			0.19927(2){{cite journal	last = Tahir-Kheli
simple cubic with 2NN (same as fcc)	12	12			0.1991(1)
simple cubic with NN+4NN	12	12			0.15040(12),{{cite journal	last = Malarz
simple cubic with 3NN+4NN	14	14			0.20490(12)	0.1012133(7)
bcc NN+2NN (= sc(3,4) sc-3NN+4NN)	14	14			0.175, 0.1686,(20) 0.1759432(8)	0.0991(5), 0.1012133(7),{{cite journal
Nanotube fibers on FCC	14	14			0.1533(13){{cite journal	last = Xu
simple cubic with NN+3NN	14	14			0.1420(1){{cite journal	last = Kurzawski
simple cubic with 2NN+4NN	18	18			0.15950(12)	0.0751589(9)
simple cubic with NN+2NN	18	18			date=June 2019}} 0.1373045(5)	0.0752326(6)
fcc with NN+2NN (=sc-2NN+4NN)	18	18			0.136, 0.1361408(8)	0.0751589(9)
simple cubic with short-length correlation	6+	6+			0.126(1){{cite journal	last = Harter
simple cubic with NN+3NN+4NN	20	20			0.11920(12)	0.0624379(9)
simple cubic with 2NN+3NN	20	20			0.1036(1)	0.0629283(7)
simple cubic with NN+2NN+4NN	24	24			0.11440(12)	0.0533056(6)
simple cubic with 2NN+3NN+4NN	26	26			0.11330(12)	0.0474609(9)
simple cubic with NN+2NN+3NN	26	26			0.097, 0.0976(1), 0.0976445(10), 0.0976444(6)	0.0497080(10)
bcc with NN+2NN+3NN	26	26			0.095, 0.0959084(6)	0.0492760(10)
simple cubic with NN+2NN+3NN+4NN	32	32			0.10000(12), 0.0801171(9)	0.0392312(8)
fcc with NN+2NN+3NN	42	42			0.061, 0.0610(5),{{cite journal	last = Gawron
fcc with NN+2NN+3NN+4NN	54	54			0.0500(5)
sc-1,2,3,4,5 simple cubic with NN+2NN+3NN+4NN+5NN	56	56			0.0461815(5)	0.0210977(7)
sc-1,...,6 (2x2x2 cube )	80	80			0.0337049(9), 0.03373(13)	0.0143950(10)
sc-1,...,7	92	92			0.0290800(10)	0.0123632(8)
sc-1,...,8	122	122			0.0218686(6)	0.0091337(7)
sc-1,...,9	146	146			0.0184060(10)	0.0075532(8)
sc-1,...,10	170	170				0.0064352(8)
sc-1,...,11	178	178				0.0061312(8)
sc-1,...,12	202	202				0.0053670(10)
sc-1,...,13	250	250				0.0042962(8)
3x3x3 cube	274	274			φc= 0.76564(1), pc = 0.0098417(7), 0.009854(6)
4x4x4 cube	636	636			φc=0.76362(1), pc = 0.0042050(2), 0.004217(3)
5x5x5 cube	1214	1250			φc=0.76044(2), pc = 0.0021885(2), 0.002185(4)
6x6x6 cube	2056	2056			0.001289(2)
::

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 {{cite journal | last1=Sykes | first1=M. F. | last2=Rehr | first2=J. J. | last3=Glen | first3=Maureen | title = A note on the percolation probabilities of pairs of closely similar lattices | journal = Mathematical Proceedings of the Cambridge Philosophical Society | volume = 76 | year = 1996 | pages = 389–392 | doi =10.1017/S0305004100049021| s2cid = 96528423

::data[format=table]

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

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.

::data[format=table]

Lattice	\alpha	d	Site percolation threshold
cubic	0.05	1.0	0.60254(3)
	0.1	1.00625	0.58688(4)
	0.15	1.025	0.55075(2)
	0.175	1.05	0.50645(5)
	0.2	1.1	0.44342(3)
::

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

::data[format=table]

System	k	z	Site coverage φc	Site percolation threshold pc
1 x 2 dimer, cubic lattice	2	56	0.24542	0.045847(2)
1 x 3 trimer, cubic lattice	3	104	0.19578	0.023919(9)
1 x 4 stick, cubic lattice	4	164	0.16055	0.014478(7)
1 x 5 stick, cubic lattice	5	236	0.13488	0.009613(8)
1 x 6 stick, cubic lattice	6	320	0.11569	0.006807(2)
2 x 2 plaquette, cubic lattice	2		0.22710	0.021238(2)
3 x 3 plaquette, cubic lattice	3		0.18686	0.007632(5)
4 x 4 plaquette, cubic lattice	4		0.16159	0.003665(3)
5 x 5 plaquette, cubic lattice	5		0.14316	0.002058(5)
6 x 6 plaquette, cubic lattice	6		0.12900	0.001278(5)
::

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

::data[format=table]

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

Thresholds for 3D continuum models

All overlapping except for jammed spheres and polymer matrix. ::data[format=table]

System	Φc	ηc
Spheres of radius r	0.289,{{cite journal	last = Holcomb
Oblate ellipsoids with major radius r and aspect ratio	0.2831{{cite journal	last = Garboczi
Prolate ellipsoids with minor radius r and aspect ratio	0.2757, 0.2795, 0.2763	0.3278
Oblate ellipsoids with major radius r and aspect ratio 2	0.2537, 0.2629, 0.254	0.3050
Prolate ellipsoids with minor radius r and aspect ratio 2	0.2537, 0.2618, 0.25(2),{{cite journal	last = Yi
Oblate ellipsoids with major radius r and aspect ratio 3	0.2289	0.2599
Prolate ellipsoids with minor radius r and aspect ratio 3	0.2033, 0.2244, 0.20(2)	0.2541, 0.22(3)
Oblate ellipsoids with major radius r and aspect ratio 4	0.2003	0.2235
Prolate ellipsoids with minor radius r and aspect ratio 4	0.1901, 0.16(2)	0.2108, 0.17(3)
Oblate ellipsoids with major radius r and aspect ratio 5	0.1757	0.1932
Prolate ellipsoids with minor radius r and aspect ratio 5	0.1627, 0.13(2)	0.1776, 0.15(2)
Oblate ellipsoids with major radius r and aspect ratio 10	0.0895, 0.1058	0.1118
Prolate ellipsoids with minor radius r and aspect ratio 10	0.0724, 0.08703, 0.07(2)	0.09105, 0.07(2)
Oblate ellipsoids with major radius r and aspect ratio 100	0.01248	0.01256
Prolate ellipsoids with minor radius r and aspect ratio 100	0.006949	0.006973
Oblate ellipsoids with major radius r and aspect ratio 1000	0.001275	0.001276
Oblate ellipsoids with major radius r and aspect ratio 2000	0.000637	0.000637
Spherocylinders with H/D = 1	0.2439(2)
Spherocylinders with H/D = 4	0.1345(1)
Spherocylinders with H/D = 10	0.06418(20)
Spherocylinders with H/D = 50	0.01440(8)
Spherocylinders with H/D = 100	0.007156(50)
Spherocylinders with H/D = 200	0.003724(90)
Aligned cylinders	0.2819(2)	0.3312(1){{cite journal
Aligned cubes of side \ell = 2 a	0.2773(2) 0.27727(2), 0.27730261(79)	0.3247(3), 0.3248(3), 0.32476(4) 0.324766(1)
Randomly oriented icosahedra		0.3030(5)
Randomly oriented dodecahedra		0.2949(5)
Randomly oriented octahedra		0.2514(6)
Randomly oriented cubes of side \ell = 2 a	0.2168(2) 0.2174,	0.2444(3), 0.2443(5){{cite journal
Randomly oriented tetrahedra		0.1701(7)
Randomly oriented disks of radius r (in 3D)		0.9614(5){{cite journal
Randomly oriented square plates of side \sqrt{\pi} r		0.8647(6)
Randomly oriented triangular plates of side \sqrt{2 \pi} /3^{1/4} r		0.7295(6)
Jammed spheres (average z = 6)	0.183(3), 0.1990,	last = Ziff
::

\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.

For more ellipsoid percolation values see.

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.

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

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

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. ::data[format=table]

System	Φc	ηc
Voids around disks of radius r		22.86(2)
Voids around randomly oriented tetrahedra	0.0605(6)
Voids around oblate ellipsoids of major radius r and aspect ratio 32	0.5308(7)	0.6333
Voids around oblate ellipsoids of major radius r and aspect ratio 16	0.3248(5)	1.125
Voids around oblate ellipsoids of major radius r and aspect ratio 10		1.542(1)
Voids around oblate ellipsoids of major radius r and aspect ratio 8	0.1615(4)	1.823
Voids around oblate ellipsoids of major radius r and aspect ratio 4	0.0711(2)	2.643, 2.618(5)
Voids around oblate ellipsoids of major radius r and aspect ratio 2		3.239(4)
Voids around prolate ellipsoids of aspect ratio 8	0.0415(7)
Voids around prolate ellipsoids of aspect ratio 6	0.0397(7)
Voids around prolate ellipsoids of aspect ratio 4	0.0376(7)
Voids around prolate ellipsoids of aspect ratio 3	0.03503(50)
Voids around prolate ellipsoids of aspect ratio 2	0.0323(5)
Voids around aligned square prisms of aspect ratio 2	0.0379(5)
Voids around randomly oriented square prisms of aspect ratio 20	0.0534(4)
Voids around randomly oriented square prisms of aspect ratio 15	0.0535(4)
Voids around randomly oriented square prisms of aspect ratio 10	0.0524(5)
Voids around randomly oriented square prisms of aspect ratio 8	0.0523(6)
Voids around randomly oriented square prisms of aspect ratio 7	0.0519(3)
Voids around randomly oriented square prisms of aspect ratio 6	0.0519(5)
Voids around randomly oriented square prisms of aspect ratio 5	0.0515(7)
Voids around randomly oriented square prisms of aspect ratio 4	0.0505(7)
Voids around randomly oriented square prisms of aspect ratio 3	0.0485(11)
Voids around randomly oriented square prisms of aspect ratio 5/2	0.0483(8)
Voids around randomly oriented square prisms of aspect ratio 2	0.0465(7)
Voids around randomly oriented square prisms of aspect ratio 3/2	0.0461(14)
Voids around hemispheres	0.0455(6)
Voids around aligned tetrahedra	0.0605(6)
Voids around randomly oriented tetrahedra	0.0605(6)
Voids around aligned cubes	0.036(1), 0.0381(3)
Voids around randomly oriented cubes	0.0452(6), 0.0449(5)
Voids around aligned octahedra	0.0407(3)
Voids around randomly oriented octahedra	0.0398(5)
Voids around aligned dodecahedra	0.0356(3)
Voids around randomly oriented dodecahedra	0.0360(3)
Voids around aligned icosahedra	0.0346(3)
Voids around randomly oriented icosahedra	0.0336(7)
Voids around spheres	0.034(7),{{cite journal	last = Kertesz
::

Thresholds on 3D random and quasi-lattices

::data[format=table]

Lattice	\overline z	Site percolation threshold	Bond percolation threshold
Contact network of packed spheres	6	0.310(5),	last = Powell
Random-plane tessellation, dual	6	0.290(7)	author2=E. Dumonteil
Icosahedral Penrose	6	0.285{{cite journal	last = Zakalyukin
Penrose w/2 diagonals	6.764	0.271	0.207
Penrose w/8 diagonals	12.764	0.188	0.111
Voronoi network	15.54	0.1453(20)	last = Jerauld
::

Thresholds for other 3D models

::data[format=table]

Lattice	z	\overline z	Site percolation threshold	Critical coverage fraction \phi_c	Bond percolation threshold
Drilling percolation, simple cubic lattice*	6	6	0.6345(3),	last = Kantor	first = Yacov
Drill in z direction on cubic lattice, remove single sites	6	6	0.592746 (columns), 0.4695(10) (sites)	0.2784
Random tube model, simple cubic lattice†					0.231456(6)
Pac-Man percolation, simple cubic lattice					0.139(6)
::

^* 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.

Thresholds in different dimensional spaces

Continuum models in higher dimensions

::data[format=table]

d	System	Φc	ηc
4	Overlapping hyperspheres	0.1223(4)	0.1300(13), 0.1304(5), 0.1210268(19)
4	Aligned hypercubes	0.1132(5), 0.1132348(17)	0.1201(6)
4	Voids around hyperspheres	0.00211(2)	6.161(10) 6.248(2),
5	Overlapping hyperspheres		0.0544(6), 0.05443(7), 0.0522524(69)
5	Aligned hypercubes	0.04900(7), 0.0481621(13)	0.05024(7)
5	Voids around hyperspheres	1.26(6)x10−4	8.98(4), 9.170(8)
6	Overlapping hyperspheres		0.02391(31), 0.02339(5)
6	Aligned hypercubes	0.02082(8), 0.0213479(10)	0.02104(8)
6	Voids around hyperspheres	8.0(6)x10−6	11.74(8), 12.24(2),
7	Overlapping hyperspheres		0.01102(16), 0.01051(3)
7	Aligned hypercubes	0.00999(5), 0.0097754(31)	0.01004(5)
7	Voids around hyperspheres		15.46(5)
8	Overlapping hyperspheres		0.00516(8), 0.004904(6)
8	Aligned hypercubes		0.004498(5)
8	Voids around hyperspheres		18.64(8)
9	Overlapping hyperspheres		0.002353(4)
9	Aligned hypercubes		0.002166(4)
9	Voids around hyperspheres		22.1(4)
10	Overlapping hyperspheres		0.001138(3)
10	Aligned hypercubes		0.001058(4)
11	Overlapping hyperspheres		0.0005530(3)
11	Aligned hypercubes		0.0005160(3)
::

\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

::data[format=table]

d	z	Site thresholds	Bond thresholds
4	8	0.198(1){{cite journal	last = Kirkpatrick
5	10	0.141(1),0.198(1) 0.141(3), 0.1407966(15), 0.1407966(26), 0.14079633(4)	last1=Harris
6	12	0.106(1), 0.108(3), 0.109017(2), 0.1090117(30), 0.109016661(8)	0.0943(1), 0.0942(1), 0.0942019(6), 0.09420165(2)
7	14	0.05950(5), 0.088939(20),{{cite journal	last = Stauffer
8	16	0.0752101(5), 0.075210128(1)	0.06770(5), 0.06770839(7), 0.0677084181(3)
9	18	0.0652095(3), 0.0652095348(6)	0.05950(5),{{cite journal
10	20	0.0575930(1), 0.0575929488(4)	0.05309258(4), 0.0530925842(2)
11	22	0.05158971(8), 0.0515896843(2)	0.04794969(1), 0.04794968373(8)
12	24	0.04673099(6), 0.0467309755(1)	0.04372386(1), 0.04372385825(10)
13	26	0.04271508(8), 0.04271507960(10){{Cite journal	last = Mertens
::

For thresholds on high dimensional hypercubic lattices, we have the asymptotic series expansions

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

::data[format=table]

d	lattice	z	Site thresholds	Bond thresholds
4	diamond	5	0.2978(2)	0.2715(3)
4	kagome	8	0.2715(3){{cite journal	last = van der Marck
4	bcc	16	0.1037(3)	0.074(1), 0.074212(1){{cite journal
4	fcc, D4, hypercubic 2NN	24	0.0842(3), 0.08410(23), 0.0842001(11)	0.049(1), 0.049517(1), 0.0495193(8)
4	hypercubic NN+2NN	32	0.06190(23), 0.0617731(19){{cite journal	last = Zhao
4	hypercubic 3NN	32	0.04540(23)
4	hypercubic NN+3NN	40	0.04000(23)	0.0271892(22)
4	hypercubic 2NN+3NN	56	0.03310(23)	0.0194075(15)
4	hypercubic NN+2NN+3NN	64	0.03190(23), 0.0319407(13)	0.0171036(11)
4	hypercubic NN+2NN+3NN+4NN	88	0.0231538(12)	0.0122088(8)
4	hypercubic NN+...+5NN	136	0.0147918(12)	0.0077389(9)
4	hypercubic NN+...+6NN	232	0.0088400(10)	0.0044656(11)
4	hypercubic NN+...+7NN	296	0.0070006(6)	0.0034812(7)
4	hypercubic NN+...+8NN	320	0.0064681(9)	0.0032143(8)
4	hypercubic NN+...+9NN	424	0.0048301(9)	0.0024117(7)
5	diamond	6	0.2252(3)	0.2084(4)
5	kagome	10	0.2084(4)	0.130(2)
5	bcc	32	0.0446(4)	0.033(1)
5	fcc, D5, hypercubic 2NN	40	0.0431(3), 0.0435913(6)	0.026(2), 0.0271813(2)
5	hypercubic NN+2NN	50	0.0334(2){{cite journal	last = Löbl
6	diamond	7	0.1799(5)	0.1677(7)
6	kagome	12	0.1677(7)
6	fcc, D6	60	0.0252(5), 0.02602674(12)	0.01741556(5)
6	bcc	64	0.0199(5)
6	E6	72	0.02194021(14)	0.01443205(8)
7	fcc, D7	84	0.01716730(5)	0.012217868(13)
7	E7	126	0.01162306(4)	0.00808368(2)
8	fcc, D8	112	0.01215392(4)	0.009081804(6)
8	E8	240	0.00576991(2)	0.004202070(2)
9	fcc, D9	144	0.00905870(2)	0.007028457(3)
9	\Lambda_9	272	0.00480839(2)	0.0037006865(11)
10	fcc, D10	180	0.007016353(9)	0.005605579(6)
11	fcc, D11	220	0.005597592(4)	0.004577155(3)
12	fcc, D12	264	0.004571339(4)	0.003808960(2)
13	fcc, D13	312	0.003804565(3)	0.0032197013(14)
::

Thresholds in one-dimensional long-range percolation

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a4/LR_1d_clusters_wiki.png" caption="Long-range bond percolation model. The lines represent the possible bonds with width decreasing as the connection probability decreases (left panel). An instance of the model together with the clusters generated (right panel)."] ::

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 \sigma0. Percolation occurs at a critical value C_c for \sigma. The numerically determined percolation thresholds are given by:

::data[format=table title=""]

\sigma	C_c
last1=Gori	first1=G.
The dotted line is the rigorous lower bound.
0.1	0.047685(8)
0.2	0.093211(16)
0.3	0.140546(17)
0.4	0.193471(15)
0.5	0.25482(5)
0.6	0.327098(6)
0.7	0.413752(14)
0.8	0.521001(14)
0.9	0.66408(7)
::

Thresholds on hyperbolic, hierarchical, and tree lattices

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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/fe/TriangularHyperbolic.jpg" caption="Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk (red bonds). Green bonds show dual-clusters on the {7,3} lattice{{cite journal"] ::

::data[format=table]

Lattice	z	Site percolation threshold	Bond percolation threshold
\overline z
			Lower
{3,7} hyperbolic	7	7	0.26931171(7), 0.20
{3,8} hyperbolic	8	8	0.20878618(9)
{3,9} hyperbolic	9	9	0.1715770(1)
{4,5} hyperbolic	5	5	0.29890539(6)
{4,6} hyperbolic	6	6	0.22330172(3)
{4,7} hyperbolic	7	7	0.17979594(1)
{4,8} hyperbolic	8	8	0.151035321(9)
{4,9} hyperbolic	8	8	0.13045681(3)
{5,5} hyperbolic	5	5	0.26186660(5)
{7,3} hyperbolic	3	3	0.54710885(10)
{∞,3} Cayley tree	3	3
Enhanced binary tree (EBT)
Enhanced binary tree dual
Non-Planar Hanoi Network (HN-NP)
Cayley tree with grandparents		8
::

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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/f9/(1+1)D_Kagome_Lattice.png" caption="(1+1)d Kagome Lattice"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/b5/(1+1)D_Square_Lattice.png" caption="(1+1)d Square Lattice"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/fa/(1+1)D_Triangular_Lattice.png" caption="(1+1)d Triangular Lattice"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/d/d7/(2+1)D_SC_Lattice.png" caption="(2+1)d SC Lattice"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/3/3d/(2+1)D_BCC_Lattice.png" caption="(2+1)d BCC Lattice"] ::

::data[format=table]

Lattice	z	Site percolation threshold	Bond percolation threshold
(1+1)-d honeycomb	1.5	0.8399316(2), 0.839933(5),{{cite journal	last = Jensen
(1+1)-d kagome	2	0.7369317(2), 0.73693182(4)	0.6589689(2), 0.65896910(8)
(1+1)-d square, diagonal	2	0.705489(4),{{cite journal	last = Essam
(1+1)-d triangular	3	0.595646(3), 0.5956468(5), 0.5956470(3)	0.478018(2), 0.478025(1), 0.4780250(4) 0.479(3)
(2+1)-d simple cubic, diagonal planes	3	0.43531(1), 0.43531411(10)	0.382223(7), 0.38222462(6) 0.383(3)
(2+1)-d square nn (= bcc)	4	0.3445736(3),{{cite journal	last = Grassberger
(2+1)-d fcc			0.199(2))
(3+1)-d hypercubic, diagonal	4	0.3025(10),{{cite journal	last = Adler
(3+1)-d cubic, nn	6	0.2081040(4)	0.1774970(5)
(3+1)-d bcc	8	0.160950(30), 0.16096128(3)	0.13237417(2)
(4+1)-d hypercubic, diagonal	5	0.23104686(3)	0.20791816(2), 0.2085(2){{cite journal
(4+1)-d hypercubic, nn	8	0.1461593(2), 0.1461582(3){{cite journal	last = Grassberger
(4+1)-d bcc	16	0.075582(17),{{cite journal	last = Lübeck
(5+1)-d hypercubic, diagonal	6	0.18651358(2)	0.170615155(5), 0.1714(1)
(5+1)-d hypercubic, nn	10	0.1123373(2)	0.1016796(5)
(5+1)-d hypercubic bcc	32	0.035967(23), 0.035972540(3)	0.0314566318(5)
(6+1)-d hypercubic, diagonal	7	0.15654718(1)	0.145089946(3), 0.1458
(6+1)-d hypercubic, nn	12	0.0913087(2)	0.0841997(14)
(6+1)-d hypercubic bcc	64	0.017333051(2)	0.01565938296(10)
(7+1)-d hypercubic, diagonal	8	0.135004176(10)	0.126387509(3), 0.1270(1)
(7+1)-d hypercubic,nn	14	0.07699336(7)	0.07195(5)
(7+1)-d bcc	128	0.008 432 989(2)	0.007 818 371 82(6)
::

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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/78/DirectedPercolationNeighborhoods.png" caption="Directed Percolation neighborhoods for extended range. Upper: z = 2, 4, 6; lower: z = 3, 5"] ::

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

::data[format=table]

Lattice	z	Bond percolation threshold
(1+1)-d square	3	0.4395(3),{{cite journal
(1+1)-d square	5	0.2249(3)
(1+1)-d square	7	0.1470(2)
(1+1)-d square	9	0.1081(2)
(1+1)-d square	11	0.0851(2)
(1+1)-d square	13	0.0701(2)
(1+1)-d tilted sq	2	0.6447(2)
(1+1)-d tilted sq	4	0.3272(2)
(1+1)-d tilted sq	6	0.2121(3)
(1+1)-d tilted sq	8	0.1553(3)
(1+1)-d tilted sq	10	0.1220(2)
(1+1)-d tilted sq	12	0.0999(2)
::

For large z, pc ~ 1/z

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

::data[format=table]

Lattice	z	ps	pb	P0	P2	P3
(1+1)-d square	3	0.644701	1	0.126237	0.229062	0.415639
		0.7	0.93585	0.148376	0.196529	0.458567
		0.75	0.88565	0.169703	0.166059	0.498178
		0.8	0.84135	0.192304	0.134616	0.538464
		0.85	0.80190	0.216143	0.102242	0.579373
		0.9	0.76645	0.241215	0.068981	0.620825
		0.95	0.73450	0.267336	0.034889	0.662886
		1	0.705489	0.294511	0	0.705489
::

Isotropic/Directed Percolation

Here we have a cross between ordinary bond percolation (OP) and directed percolation (DP). 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:

::data[format=table]

Lattice	d	z	pd	pc	p↓	p↑
(1+1)-d DP	2	2	1	0.644700185(5)	0.644700185(5)	0
diagonal square	2	4	0.8	0.768708(1)	0.614966	0.153742
diagonal square	2	4	0.6	0.929668(3)	0.557801s	0.371867
2d ordinary perc.	2	4	0.5	1.0	0.5	0.5
(2+1)-d diagonal DP	3	3	1	0.38222462(6)	0.38222462	0
diagonal cubic	3	6	0.8	0.430941(2)	0.34475282	0.086188
diagonal cubic	3	6	0.6	0.481310(2)	0.288786	0.192524
3d Ordinary perc.	3	6	0.5	0.49762364(20)	0.24881182	0.24881182
::

Exact critical manifolds of inhomogeneous systems

Inhomogeneous triangular lattice bond percolation

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

Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation

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{{cite journal | last = Wu | first = F. Y. | title = Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices I: Closed-form expressions | journal = Physical Review E | volume = 81 | issue = 6 | year = 2010 | article-number = 061110 | doi = 10.1103/PhysRevE.81.061110 | pmid = 20866381 | arxiv = 0911.2514|bibcode = 2010PhRvE..81f1110W | s2cid = 31590247

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{{cite journal | last = Damavandi | first = Ojan Khatib | author2 = Robert M. Ziff | title = Percolation on hypergraphs with four-edges | journal = J. Phys. A: Math. Theor. | volume = 48 | issue = 40 | year = 2015 | article-number = 405004 | doi = 10.1088/1751-8113/48/40/405004 | arxiv = 1506.06125|bibcode = 2015JPhA...48N5004K | s2cid = 118481075

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

Inhomogeneous martini lattice, bond percolation

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{{cite journal | last = Wu | first = F. Y. | title =New Critical Frontiers for the Potts and Percolation Models | journal = Physical Review Letters | volume = 96 | issue = 9 | year = 2006 | article-number = 090602 | doi = 10.1103/PhysRevLett.96.090602|arxiv = cond-mat/0601150 |bibcode = 2006PhRvL..96i0602W | pmid=16606250| citeseerx = 10.1.1.241.6346 | s2cid = 15182833

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{{cite journal | last = Ziff | first = R. M. |author2=C. R. Scullard |author3=J. C. Wierman |author4=M. R. A. Sedlock | title = The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices | journal = Journal of Physics A | volume = 45 | issue = 49 | year = 2012 | article-number = 494005 | doi = 10.1088/1751-8113/45/49/494005 |arxiv = 1210.6609 |bibcode = 2012JPhA...45W4005Z | s2cid = 2121370

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

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.{{cite journal | last = Thorpe | first = M. | author2 = D. Jacobs | author3 = M. Chubynsky | author4 = J. Phillips | year = 2000 | title = Self-organization in network glasses | journal = Journal of Non-Crystalline Solids | volume = 266–269 | issue = | pages = 859–866 | doi = 10.1016/S0022-3093(99)00856-X | arxiv = | last = Lu | first = Mingzhong | author2 = Yufeng Song | author3 = Qiyuan Shi | author4 = Ming Li | author5 = Youjin Deng | year = 2026 | title = High-precision Dynamic Monte Carlo Study of Rigidity Percolation | journal = Arxiv preprint | volume = | issue = | article-number = 2601.21399 | doi = 10.48550/arXiv.2601.21399 | arxiv = 2601.21399v1

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.

Results:

Bond threshold, triangular lattice: pc = 0.6602(3) 0.6602778(10)

Site percolation, triangular lattice pc = 0.69755(3), 0.6975(3)

Correlation-length exponent: ν = 1.16(3), 1.19(1), 1.21(6), 1/ν = 0.850(3)

𝛼 = -0.48(5)

β = 0.175(2)

Fractal dimension df = 1.86(2). 1.853(5), 1.850(2)

Backbone fractal dimension db = 1.80(3), 1.78(2)

Arbibi Sahimi 93: 2d bond tri: p„=0.641(1), site: p„=0.713(2).

Chubynsky and Thorpe 07. 3d: bond fcc, pc = 0.495. bcc: pc = 0.7485

Javerzam arXiv:2301.07614v2. 2d hull fractal dimension : df = 1.355(10)

Roux, Hansen 88: central force elastic network: p* = 0.642(2), flv = 3.0(4) , glv = 0.97(2) ;

Arababi, Sahimi 88: . 3d bond cubic elastic network pc = 0.2492,

Sahimi, Goddard 85 bond triangular p„=0.65''

Lemieux, Breton, Tremblay 85 Pcen = 0.649, f = 1.4

Feng Sen 84 Pcen = 0.58, f = 2.4 ± 0.4.

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