Percolation threshold

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, 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 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 or bonds 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.
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 will be a step function at the threshold value p_c. For finite large systems, P is a constant whose value depends upon the shape of the system; for the square system discussed above, P= exactly for any lattice by a simple symmetry argument.
There are other signatures of the critical threshold. For example, the size distribution drops off as a power-law for large s at the threshold, n_s ~ s^−τ, where τ is a dimension-dependent percolation critical exponents. For an infinite system, the critical threshold corresponds to the first point 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. 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 all have site thresholds of, and self-dual lattices have bond thresholds of.
The notation such as 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 signifies 0.729724 ± 0.000003, and 0.74042195 signifies 0.74042195 ± 0.00000080. The error bars variously represent one or two standard deviations in net error, or an empirical confidence interval, depending upon the source.

Percolation on networks

For a random tree-like network without degree-degree correlation, it can be shown that such network can have a giant component, and the percolation threshold is given by
Where is the generating function corresponding to the excess degree distribution, is the average degree of the network and is the second moment of the degree distribution. So, for example, for an ER network, since the degree distribution is a Poisson distribution, where the threshold is at.
In networks with low clustering,, the critical point gets scaled by such that:
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

Lattice	z		Site percolation threshold	Bond percolation threshold
3-12 or super-kagome,	3	3	0.807900764... =	0.74042195, 0.74042077, 0.740420800, 0.7404207988509, 0.740420798850811610,
cross, truncated trihexagonal	3	3	0.746, 0.750, 0.747806, 0.7478008	0.6937314, 0.69373383, 0.693733124922
square octagon, bathroom tile, 4-8, truncated square	3	-	0.729, 0.729724, 0.7297232	0.6768, 0.67680232, 0.6768031269, 0.6768031243900113,
honeycomb	3	3	0.6962, 0.697040230, 0.6970402, 0.6970413, 0.697043,	0.652703645... = 1-2 sin, 1+ p³-3p²=0
kagome	4	4	0.652703645... = 1 − 2 sin	0.5244053, 0.52440516, 0.52440499, 0.524404978, 0.52440572..., 0.52440500, 0.524404999173, 0.524404999167439 0.52440499916744820
ruby, rhombitrihexagonal	4	4	0.620, 0.621819, 0.62181207	0.52483258, 0.5248311, 0.524831461573
square	4	4	0.59274, 0.59274605079210, 0.59274601, 0.59274605095, 0.59274621, 0.592746050786, 0.5927460507896, 0.59274621, 0.59274598, 0.59274605, 0.593, 0.591, 0.569, 0.59274
snub hexagonal, maple leaf	5	5	0.579 0.579498	0.43430621, 0.43432764, 0.4343283172240,
snub square, puzzle	5	5	0.550, 0.550806	0.41413743, 0.4141378476, 0.4141378565917,
frieze, elongated triangular	5	5	0.549, 0.550213, 0.5502	0.4196, 0.41964191, 0.41964044, 0.41964035886369
triangular	6	6		0.347296355... = 2 sin, 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, etc. Equivalent to square-2N+3N+4N, sq. tri = triangular, hc = honeycomb.

Lattice	z	Site percolation threshold	Bond percolation threshold
sq-1, sq-2, sq-3, sq-5	4	0.5927...
sq-1,2, sq-2,3, sq-3,5... 3x3 square	8	0.407...	0.25036834, 0.2503685, 0.25036840
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-p_c
tri-1,2: triangular-NN+2NN	12	0.295, 0.289, 0.290258
tri-2,3: triangular-2NN+3NN	12	0.232020, 0.232020
sq-4: square-4NN	8	0.270...
sq-1,5: square-NN+5NN	8	0.277
sq-1,2,3: square-NN+2NN+3NN	12	0.292, 0.290 0.289, 0.288, 0.2891226	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, 0.1924428
hc-2,4: honeycomb-2NN+4NN	12	0.2374	-
tri-1,3: triangular-NN+3NN	12	0.264539
tri-1,2,3: triangular-NN+2NN+3NN	18	0.225, 0.215, 0.215459 0.2154657
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,
sq-1,2,3,4: NN+2NN+3NN+4NN	20	0.19671, 0.196, 0.196724, 0.1967293	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	24	0.164, 0.164, 0.1647124
sq-1,2,3,4,6: NN+2NN+3NN+4NN+6NN	24	0.16134,
tri-1,4,5: NN+4NN+5NN	24	0.131660
sq-1,...,6: NN+...+6NN	28	0.142, 0.1432551	0.0558493
tri-2,3,4,5: 2NN+3NN+4NN+5NN	30	0.117460 0.135823
tri-1,2,3,4,5: NN+2NN+3NN+4NN+5NN	36	0.115, 0.115740, 0.1157399
sq-1,...,7: NN+...+7NN	36	0.113, 0.1153481	0.04169608
sq lat, diamond boundary: dist. ≤ 4	40	0.105
sq-1,...,8: NN+..+8NN	44	0.095, 0.095765, 0.09580, 0.0957661
sq-1,...,9: NN+..+9NN	48	0.086	0.02974268
sq-1,...,11: NN+...+11NN	60		0.02301190
sq-1,...,23	148		0.008342595
sq-1,...,32: NN+...+32NN	224		0.0053050415
sq-1,...,86: NN+...+86NN	708		0.001557644
sq-1,...,141: NN+...+141NN	1224		0.000880188
sq-1,...,185: NN+...+185NN	1652		0.000645458
sq-1,...,317: NN+...+317NN	3000		0.000349601
sq-1,...,413: NN+...+413NN	4016		0.0002594722
sq lat, diamond boundary: dist. ≤ 6	84	0.049
sq lat, diamond boundary: dist. ≤ 8	144	0.028
sq lat, diamond boundary: dist. ≤ 10	220	0.019
2x2 touching lattice squares*	20	φ_c = 0.58365, p_c = 0.196724, 0.19671,	-
3x3 touching lattice squares* )	44	φ_c = 0.59586, p_c = 0.095765, 0.09580
4x4 touching lattice squares*	76	φ_c = 0.60648, p_c = 0.0566227, 0.05665,
5x5 touching lattice squares*	116	φ_c = 0.61467, p_c = 0.037428, 0.03745,
6x6 touching lattice squares*	220	p_c = 0.02663,
10x10 touching lattice squares*	436	φ_c = 0.63609, p_c = 0.0100576
within 11 x 11 square	120		0.01048079
within 15 x 15 square	224		0.005287692
20x20 touching lattice squares*	1676	φ_c = 0.65006, p_c = 0.0026215
within 31 x 31 square	960		0.001131082
100x100 touching lattice squares*	40396	φ_c = 0.66318, p_c = 0.000108815
1000x1000 touching lattice squares*	4003996	φ_c = 0.66639, p_c = 1.09778E-06

Here NN = nearest neighbor, 2NN = second nearest neighbor, 3NN = third nearest neighbor, etc. These are also called 2N, 3N, 4N respectively in some papers.

For overlapping or touching squares, given here is the net fraction of sites occupied similar to the 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 with. The 3×3 square corresponds to sq-1,2,3,4,5,6,7,8 with z=44 and. The value of z for a k x k square is ²-5.

2D distorted lattices

Here, one distorts a regular lattice of unit spacing by moving vertices uniformly within the box, and considers percolation when sites are within Euclidean distance of each other.

Lattice			Site percolation threshold	Bond percolation threshold
square	0.2	1.1	0.8025
	0.2	1.2	0.6667
	0.1	1.1	0.6619

Overlapping shapes on 2D lattices

Site threshold is number of overlapping objects per lattice site. k is the length. Overlapping squares are shown in the complex neighborhood section. Here z is the coordination number to k-mers of either orientation, with for sticks.

System	k	z	Site coverage φ_c	Site percolation threshold p_c
1 x 2 dimer, square lattice	2	22	0.54691 0.5483	0.17956 0.18019
1 x 2 aligned dimer, square lattice	2	14	0.5715	0.3454
1 x 3 trimer, square lattice	3	37	0.49898 0.50004	0.10880 0.1093
1 x 4 stick, square lattice	4	54	0.45761	0.07362
1 x 5 stick, square lattice	5	73	0.42241	0.05341
1 x 6 stick, square lattice	6	94	0.39219	0.04063

The coverage is calculated from by for sticks, because there are sites where a stick will cause an overlap with a given site.
For aligned sticks:

AB percolation and colored percolation in 2D

In AB percolation, a 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 colors with equal probability, and connection is made along bonds between neighbors of different colors.

Lattice	z		Site percolation threshold
triangular AB	6	6	0.2145, 0.21524, 0.21564
AB on square-covering lattice	6	6
square three-color	4	4	0.80745
square four-color	4	4	0.73415
square five-color	4	4	0.69864
square six-color	4	4	0.67751
triangular two-color	6	6	0.72890
triangular three-color	6	6	0.63005
triangular four-color	6	6	0.59092
triangular five-color	6	6	0.56991
triangular six-color	6	6	0.55679

Site-bond percolation in 2D

Site bond percolation. Here is the site occupation probability and 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 = 0, and some specific critical pairs are listed below.
Square lattice:

Lattice	z		Site percolation threshold	Bond percolation threshold
square	4	4	0.615185	0.95
			0.667280	0.85
			0.732100	0.75
			0.75	0.726195
			0.815560	0.65
			0.85	0.615810
			0.95	0.533620

Honeycomb lattice:

Lattice	z		Site percolation threshold	Bond percolation threshold
honeycomb	3	3	0.7275	0.95
			0. 0.7610	0.90
			0.7986	0.85
			0.80	0.8481
			0.8401	0.80
			0.85	0.7890
			0.90	0.7377
			0.95	0.6926

Kagome lattice:

Lattice	z		Site percolation threshold	Bond percolation threshold
kagome	4	4	0.6711, 0.67097	0.95
			0.6914, 0.69210	0.90
			0.7162, 0.71626	0.85
			0.7428, 0.74339	0.80
			0.75	0.7894
			0.7757, 0.77556	0.75
			0.80	0.7152
			0.81206	0.70
			0.85	0.6556
			0.85519	0.65
			0.90	0.6046
			0.90546	0.60
			0.95	0.5615
			0.96604	0.55
			0.9854	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

Lattice	z		Threshold	Notes
honeycomb	3	3	, When equal: p_s = p_b = 0.82199	approximate formula, p_s = site prob., p_b = bond prob., p_bc = 1 − 2 sin, exact at p_s=1, p_b=p_bc.

Archimedean duals (Laves lattices)

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

Lattice	z		Site percolation threshold	Bond percolation threshold
Cairo pentagonal D=+	3,4	3	0.6501834, 0.650184	0.585863... = 1 − p_c^bond
Pentagonal D=+	3,4	3	0.6470471, 0.647084, 0.6471	0.580358... = 1 − p_c^bond, 0.5800
D=+	3,6	3	0.639447	0.565694... = 1 − p_c^bond
dice, rhombille tiling D = +	3,6	4	0.5851, 0.585040	0.475595... = 1 − p_c^bond
ruby dual D = + +	3,4,6	4	0.582410	0.475167... = 1 − p_c^bond
union jack, tetrakis square tiling D = +	4,8	6		0.323197... = 1 − p_c^bond
bisected hexagon, cross dual D= ++	4,6,12	6		0.306266... = 1 − p_c^bond
asanoha D=+	3,12	6		0.259579... = 1 − p_c^bond

2-uniform lattices

Top 3 lattices: #13 #12 #36

Bottom 3 lattices: #34 #37 #11
Top 2 lattices: #35 #30

Bottom 2 lattices: #41 #42
Top 4 lattices: #22 #23 #21 #20

Bottom 3 lattices: #16 #17 #15
Top 2 lattices: #31 #32

Bottom lattice: #33

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

Inhomogeneous 2-uniform lattice

This figure shows something similar to the 2-uniform lattice #37, except the polygons are not all regular—there is a rectangle in the place of the two squares—and the size of the polygons is changed. This lattice is in the isoradial representation in which each polygon is inscribed in a circle of unit radius. The two squares in the 2-uniform lattice must now be represented as a single rectangle in order to satisfy the isoradial condition. The lattice is shown by black edges, and the dual lattice by red dashed lines. The green circles show the isoradial constraint on both the original and dual lattices. The yellow polygons highlight the three types of polygons on the lattice, and the pink polygons highlight the two types of polygons on the dual lattice. The lattice has vertex types +, while the dual lattice has vertex types +++. The critical point is where the longer bonds have occupation probability p = 2 sin = 0.347296... which is the bond percolation threshold on a triangular lattice, and the shorter bonds have occupation probability 1 − 2 sin = 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,, and 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.
Some other examples of generalized bow-tie lattices and the duals of the lattices :

Lattice	z		Site percolation threshold	Bond percolation threshold
martini +	3	3	0.764826..., 1 + p⁴ − 3p³ = 0	0.707107... = 1/
bow-tie	3,4	3		0.672929..., 1 − 2p³ − 2p⁴ − 2p⁵ − 7p⁶ + 18p⁷ + 11p⁸ − 35p⁹ + 21p¹⁰ − 4p¹¹ = 0
bow-tie	3,4	3		0.625457..., 1 − 2p² − 3p³ + 4p⁴ − p⁵ = 0
martini-A +	3,4	3	1/	0.625457..., 1 − 2p² − 3p³ + 4p⁴ − p⁵ = 0
bow-tie dual	3,4	3		0.595482..., 1-p_c^bond
bow-tie	3,4,6	3		0.533213..., 1 − p − 2p³ -4p⁴-4p⁵+15⁶+ 13p⁷-36p⁸+19p⁹+ p¹⁰ + p¹¹=0
martini covering/medial +	4	4	0.707107... = 1/	0.57086651
martini-B +	3, 5	4	0.618034... = 2/, 1- p² − p = 0
bow-tie dual	3,4,8	4		0.466787..., 1 − p_c^bond
bow-tie +	4,6	5	0.5472, 0.5479148	0.404518..., 1 − p − 6p² + 6p³ − p⁵ = 0
bow-tie dual	3,6,8	5		0.374543..., 1 − p_c^bond
bow-tie dual	3,6,10	5	0.547... = p_c^site	0.327071..., 1 − p_c^bond
martini dual +	3,9	6		0.292893... = 1 − 1/

Thresholds on 2D covering, medial, and matching lattices

Lattice	z		Site percolation threshold	Bond percolation threshold
covering/medial	4	4	p_c^bond = 0.693731...	0.5593140, 0.559315
covering/medial, square kagome	4	4	p_c^bond = 0.676803...	0.544798017, 0.54479793
medial	4	4		0.5247495
medial	4	4		0.51276
medial	4	4		0.512682929
medial	4	4		0.5125245984
square covering	6	6		0.3371
square matching lattice	8	8	1 − p_c^site = 0.407253...	0.25036834

Thresholds on subnet lattices

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

Lattice	z	Site percolation threshold	Bond percolation threshold
checkerboard – 2 × 2 subnet	4,3		0.596303
checkerboard – 4 × 4 subnet	4,3		0.633685
checkerboard – 8 × 8 subnet	4,3		0.642318
checkerboard – 16 × 16 subnet	4,3		0.64237
checkerboard – 32 × 32 subnet	4,3		0.64219
checkerboard – subnet	4,3		0.642216
kagome – 2 × 2 subnet = covering/medial	4	p_c^bond = 0.74042077...	0.600861966960, 0.6008624, 0.60086193
kagome – 3 × 3 subnet	4		0.6193296, 0.61933176, 0.61933044
kagome – 4 × 4 subnet	4		0.625365, 0.62536424
kagome – subnet	4		0.628961
kagome – : subnet = martini covering/medial	4	p_c^bond = 1/ = 0.707107...	0.57086648
kagome – : subnet	4,3	0.728355596425196...	0.58609776
kagome – : subnet		0.738348473943256...
kagome – : subnet		0.743548682503071...
kagome – : subnet		0.746418147634282...
kagome – : subnet			0.61091770
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
triangular – subnet	6,4		0.53993

Thresholds of random sequentially adsorbed objects

system	z	Site threshold
dimers on a honeycomb lattice	3	0.69, 0.6653
dimers on a triangular lattice	6	0.4872, 0.4873,
aligned linear dimers on a triangular lattice	6	0.5157
aligned linear 4-mers on a triangular lattice	6	0.5220
aligned linear 8-mers on a triangular lattice	6	0.5281
aligned linear 12-mers on a triangular lattice	6	0.5298
linear 16-mers on a triangular lattice	6	aligned 0.5328
linear 32-mers on a triangular lattice	6	aligned 0.5407
linear 64-mers on a triangular lattice	6	aligned 0.5455
aligned linear 80-mers on a triangular lattice	6	0.5500
aligned linear k on a triangular lattice	6	0.582
dimers and 5% impurities, triangular lattice	6	0.4832
parallel dimers on a square lattice	4	0.5863
dimers on a square lattice	4	0.5617, 0.5618, 0.562, 0.5713
linear 3-mers on a square lattice	4	0.528
3-site 120° angle, 5% impurities, triangular lattice	6	0.4574
3-site triangles, 5% impurities, triangular lattice	6	0.5222
linear trimers and 5% impurities, triangular lattice	6	0.4603
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
linear 10-mers on a square lattice	4	0.469
linear 16-mers on a square lattice	4	0.4639
linear 32-mers on a square lattice	4	0.4747

The threshold gives the fraction of sites occupied by the objects when site percolation first takes place. For longer k-mers see Ref.

Thresholds of full dimer coverings of two dimensional lattices

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

system	z	Bond threshold
Parallel covering, square lattice	6	0.381966...
Shifted covering, square lattice	6	0.347296...
Staggered covering, square lattice	6	0.376825
Random covering, square lattice	6	0.367713
Parallel covering, triangular lattice	10	0.237418...
Staggered covering, triangular lattice	10	0.237497
Random covering, triangular lattice	10	0.235340

Thresholds of polymers (random walks) on a square lattice

System is composed of ordinary random walks of length l on the square lattice.

l	z	Bond percolation
1	4	0.5
2	4	0.47697
4	4	0.44892
8	4	0.41880

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

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

Thresholds for 2D continuum models

For disks, equals the critical number of disks per unit area, measured in units of the diameter, where is the number of objects and is the system size
For disks, equals critical total disk area.
gives the number of disk centers within the circle of influence.
is the critical disk radius.
for ellipses of semi-major and semi-minor axes of a and b, respectively. Aspect ratio with.
for rectangles of dimensions and. Aspect ratio with.
for power-law distributed disks with,.
equals critical area fraction.
For disks, Ref. use where is the density of disks of radius.
equals number of objects of maximum length per unit area.
For ellipses,
For void percolation, is the critical void fraction.
For more ellipse values, see
For more rectangle values, see
Both ellipses and rectangles belong to the superellipses, with. 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

Thresholds on 2D random and quasi-lattices

*Theoretical estimate

Thresholds on 2D correlated systems

Assuming power-law correlations

lattice	α	Site percolation threshold	Bond percolation threshold
square	3	0.561406
square	2	0.550143
square	0.1	0.508

Thresholds on slabs

h is the thickness of the slab, h × ∞ × ∞. Boundary conditions refer to the top and bottom planes of the slab.

Lattice	h	z		Site percolation threshold	Bond percolation threshold
simple cubic	2	5	5	0.47424, 0.4756
bcc	2			0.4155
hcp	2			0.2828
diamond	2			0.5451
simple cubic	3			0.4264
bcc	3			0.3531
bcc	3				0.21113018
hcp	3			0.2548
diamond	3			0.5044
simple cubic	4			0.3997, 0.3998
bcc	4			0.3232
bcc	4				0.20235168
hcp	4			0.2405
diamond	4			0.4842
simple cubic	5	6	6		0.278102
simple cubic	6			0.3708
simple cubic	6	6	6		0.272380
bcc	6			0.2948
hcp	6			0.2261
diamond	6			0.4642
simple cubic	7	6	6	0.3459514	0.268459
simple cubic	8			0.3557, 0.3565
simple cubic	8	6	6		0.265615
bcc	8			0.2811
hcp	8			0.2190
diamond	8			0.4549
simple cubic	12			0.3411
bcc	12			0.2688
hcp	12			0.2117
diamond	12			0.4456
simple cubic	16			0.3219, 0.3339
bcc	16			0.2622
hcp	16			0.2086
diamond	16			0.4415
simple cubic	32			0.3219,
simple cubic	64			0.3165,
simple cubic	128			0.31398,

Percolation in 3D

Lattice	z		filling factor*	filling fraction*	Site percolation threshold	Bond percolation threshold
-a oxide	2³ 3²	2.4			0.748713	= = 0.742334
-b oxide	2³ 3²	2.4	0.233	0.174	0.745317	= = 0.739388
silicon dioxide	4,2²	2			0.638683
Modified -b	3²,2	2				0.627
-a	3	3			0.577962	0.555700
-a gyroid	3	3			0.571404	0.551060
-b	3	3			0.565442	0.546694
cubic oxide	6,2³	3.5			0.524652
bcc dual		4			0.4560	0.4031
ice Ih	4	4	π / 16 = 0.340087	0.147	0.433	0.388
diamond	4	4	π / 16 = 0.340087	0.1462332	0.4299, 0.4299870,, 0.4297 0.4301, 0.428, 0.425, 0.425, 0.436	0.3895892, 0.3893, 0.3893, 0.388, 0.3886, 0.388 0.390
diamond dual		6			0.3904	0.2350
3D kagome	6		π / 12 = 0.37024	0.1442	0.3895 =p_c for diamond dual and p_c for diamond lattice	0.2709
Bow-tie stack dual		5			0.3480	0.2853
honeycomb stack	5	5			0.3701	0.3093
octagonal stack dual	5	5			0.3840	0.3168
pentagonal stack		5			0.3394	0.2793
kagome stack	6	6	0.453450	0.1517	0.3346	0.2563
fcc dual	4²,8	5			0.3341	0.2703
simple cubic	6	6	π / 6 = 0.5235988	0.1631574	0.307, 0.307, 0.3115, 0.3116077, 0.311604, 0.311605, 0.311600, 0.3116077, 0.3116081, 0.3116080, 0.3116060, 0.3116004, 0.31160768	0.247, 0.2479, 0.2488, 0.24881182, 0.2488125, 0.2488126,
hcp dual	4⁴,8²	5			0.3101	0.2573
dice stack	5,8	6	π / 9 = 0.604600	0.1813	0.2998	0.2378
bow-tie stack	7	7			0.2822	0.2092
Stacked triangular / simple hexagonal	8	8			0.26240, 0.2625, 0.2623	0.18602, 0.1859
octagonal stack	6,10	8			0.2524	0.1752
bcc	8	8			0.243, 0.243, 0.2459615, 0.2460, 0.2464, 0.2458	0.178, 0.1795, 0.18025, 0.1802875
simple cubic with 3NN	8	8			0.2455, 0.2457
fcc, D₃	12	12	π / = 0.740480	0.147530	0.195, 0.198, 0.1998, 0.1992365, 0.19923517, 0.1994, 0.199236	0.1198, 0.1201635 0.120169
hcp	12	12	π / = 0.740480	0.147545	0.195, 0.1992555	0.1201640, 0.119
La_2−x Sr_x Cu O₄	12	12			0.19927
simple cubic with 2NN	12	12			0.1991
simple cubic with NN+4NN	12	12			0.15040, 0.1503793	0.1068263
simple cubic with 3NN+4NN	14	14			0.20490	0.1012133
bcc NN+2NN	14	14			0.175, 0.1686, 0.1759432	0.0991, 0.1012133, 0.1759432
Nanotube fibers on FCC	14	14			0.1533
simple cubic with NN+3NN	14	14			0.1420	0.0920213
simple cubic with 2NN+4NN	18	18			0.15950	0.0751589
simple cubic with NN+2NN	18	18			0.137, 0.136, 0.1372, 0.13735, 0.1373045	0.0752326
fcc with NN+2NN	18	18			0.136, 0.1361408	0.0751589
simple cubic with short-length correlation	6+	6+			0.126
simple cubic with NN+3NN+4NN	20	20			0.11920	0.0624379
simple cubic with 2NN+3NN	20	20			0.1036	0.0629283
simple cubic with NN+2NN+4NN	24	24			0.11440	0.0533056
simple cubic with 2NN+3NN+4NN	26	26			0.11330	0.0474609
simple cubic with NN+2NN+3NN	26	26			0.097, 0.0976, 0.0976445, 0.0976444	0.0497080
bcc with NN+2NN+3NN	26	26			0.095, 0.0959084	0.0492760
simple cubic with NN+2NN+3NN+4NN	32	32			0.10000, 0.0801171	0.0392312
fcc with NN+2NN+3NN	42	42			0.061, 0.0610, 0.0618842	0.0290193
fcc with NN+2NN+3NN+4NN	54	54			0.0500
sc-1,2,3,4,5 simple cubic with NN+2NN+3NN+4NN+5NN	56	56			0.0461815	0.0210977
sc-1,...,6	80	80			0.0337049, 0.03373	0.0143950
sc-1,...,7	92	92			0.0290800	0.0123632
sc-1,...,8	122	122			0.0218686	0.0091337
sc-1,...,9	146	146			0.0184060	0.0075532
sc-1,...,10	170	170				0.0064352
sc-1,...,11	178	178				0.0061312
sc-1,...,12	202	202				0.0053670
sc-1,...,13	250	250				0.0042962
3x3x3 cube	274	274			φ_c= 0.76564, p_c = 0.0098417, 0.009854
4x4x4 cube	636	636			φ_c=0.76362, p_c = 0.0042050, 0.004217
5x5x5 cube	1214	1250			φ_c=0.76044, p_c = 0.0021885, 0.002185
6x6x6 cube	2056	2056			0.001289

Filling factor = fraction of space filled by touching spheres at every lattice site. Also called Atomic Packing Factor.
Filling fraction = filling factor * p_c.
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, with edges and corners removed, with z = ³-12-9.
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

System	polymer Φ_c
percolating excluded volume of athermal polymer matrix	0.4304

3D distorted lattices

Here, one distorts a regular lattice of unit spacing by moving vertices uniformly within the cube, and considers percolation when sites are within Euclidean distance of each other.

Lattice			Site percolation threshold	Bond percolation threshold
cubic	0.05	1.0	0.60254
	0.1	1.00625	0.58688
	0.15	1.025	0.55075
	0.175	1.05	0.50645
	0.2	1.1	0.44342

Overlapping shapes on 3D lattices

Site threshold is the number of overlapping objects per lattice site. The coverage φ_c is the net fraction of sites covered, and v is the volume. 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

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

The coverage is calculated from by for sticks, and for plaquettes.

Dimer percolation in 3D

System	Site percolation threshold	Bond percolation threshold
Simple cubic		0.2555

Thresholds for 3D continuum models

All overlapping except for jammed spheres and polymer matrix.

System	Φ_c	η_c
Spheres of radius r	0.289, 0.293, 0.286, 0.295. 0.2895, 0.28955, 0.2896, 0.289573, 0.2896, 0.2854, 0.290, 0.290, 0.2895693	0.3418, 0.3438, 0.341889, 0.3360, 0.34189, 0.341935, 0.335,
Oblate ellipsoids with major radius r and aspect ratio	0.2831	0.3328
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, 0.2507	0.3035, 0.29
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	0.2541, 0.22
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	0.2108, 0.17
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	0.1776, 0.15
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	0.09105, 0.07
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
Spherocylinders with H/D = 4	0.1345
Spherocylinders with H/D = 10	0.06418
Spherocylinders with H/D = 50	0.01440
Spherocylinders with H/D = 100	0.007156
Spherocylinders with H/D = 200	0.003724
Aligned cylinders	0.2819	0.3312
Aligned cubes of side	0.2773 0.27727, 0.27730261	0.3247, 0.3248, 0.32476 0.324766
Randomly oriented icosahedra		0.3030
Randomly oriented dodecahedra		0.2949
Randomly oriented octahedra		0.2514
Randomly oriented cubes of side	0.2168 0.2174,	0.2444, 0.2443
Randomly oriented tetrahedra		0.1701
Randomly oriented disks of radius r		0.9614
Randomly oriented square plates of side		0.8647
Randomly oriented triangular plates of side		0.7295
Jammed spheres	0.183, 0.1990, see also contact network of jammed spheres below.	0.59

is the total volume, where N is the number of objects and L is the system size.
is the critical volume fraction, valid for overlapping randomly placed objects.
For disks and plates, these are effective volumes and volume fractions.
For void, 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, 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 refers to the fraction of the space occupied by the voids at the critical point, and is related to by
. is defined as in the continuum percolation section above.

System	Φ_c	η_c
Voids around disks of radius r		22.86
Voids around randomly oriented tetrahedra	0.0605
Voids around oblate ellipsoids of major radius r and aspect ratio 32	0.5308	0.6333
Voids around oblate ellipsoids of major radius r and aspect ratio 16	0.3248	1.125
Voids around oblate ellipsoids of major radius r and aspect ratio 10		1.542
Voids around oblate ellipsoids of major radius r and aspect ratio 8	0.1615	1.823
Voids around oblate ellipsoids of major radius r and aspect ratio 4	0.0711	2.643, 2.618
Voids around oblate ellipsoids of major radius r and aspect ratio 2		3.239
Voids around prolate ellipsoids of aspect ratio 8	0.0415
Voids around prolate ellipsoids of aspect ratio 6	0.0397
Voids around prolate ellipsoids of aspect ratio 4	0.0376
Voids around prolate ellipsoids of aspect ratio 3	0.03503
Voids around prolate ellipsoids of aspect ratio 2	0.0323
Voids around aligned square prisms of aspect ratio 2	0.0379
Voids around randomly oriented square prisms of aspect ratio 20	0.0534
Voids around randomly oriented square prisms of aspect ratio 15	0.0535
Voids around randomly oriented square prisms of aspect ratio 10	0.0524
Voids around randomly oriented square prisms of aspect ratio 8	0.0523
Voids around randomly oriented square prisms of aspect ratio 7	0.0519
Voids around randomly oriented square prisms of aspect ratio 6	0.0519
Voids around randomly oriented square prisms of aspect ratio 5	0.0515
Voids around randomly oriented square prisms of aspect ratio 4	0.0505
Voids around randomly oriented square prisms of aspect ratio 3	0.0485
Voids around randomly oriented square prisms of aspect ratio 5/2	0.0483
Voids around randomly oriented square prisms of aspect ratio 2	0.0465
Voids around randomly oriented square prisms of aspect ratio 3/2	0.0461
Voids around hemispheres	0.0455
Voids around aligned tetrahedra	0.0605
Voids around randomly oriented tetrahedra	0.0605
Voids around aligned cubes	0.036, 0.0381
Voids around randomly oriented cubes	0.0452, 0.0449
Voids around aligned octahedra	0.0407
Voids around randomly oriented octahedra	0.0398
Voids around aligned dodecahedra	0.0356
Voids around randomly oriented dodecahedra	0.0360
Voids around aligned icosahedra	0.0346
Voids around randomly oriented icosahedra	0.0336
Voids around spheres	0.034, 0.032, 0.030, 0.0301, 0.0294, 0.0300, 0.0317, 0.0308 0.0301, 0.0301	3.506, 3.515, 3.510

Thresholds on 3D random and quasi-lattices

Lattice	z		Site percolation threshold	Bond percolation threshold
Contact network of packed spheres		6	0.310, 0.287, 0.3116,
Random-plane tessellation, dual		6	0.290
Icosahedral Penrose		6	0.285	0.225
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	0.0822

Thresholds for other 3D models

Lattice	z		Site percolation threshold	Critical coverage fraction	Bond percolation threshold
Drilling percolation, simple cubic lattice*	6	6	0.6345, 0.6339, 0.633965	0.25480
Drill in z direction on cubic lattice, remove single sites	6	6	0.592746, 0.4695	0.2784
Random tube model, simple cubic lattice^†					0.231456
Pac-Man percolation, simple cubic lattice					0.139

In drilling percolation, the site threshold represents the fraction of columns in each direction that have not been removed, and. For the 1d drilling, we have .
^† In tube percolation, the bond threshold represents the value of the parameter such that the probability of putting a bond between neighboring vertical tube segments is, where is the overlap height of two adjacent tube segments.

Thresholds in different dimensional spaces

Continuum models in higher dimensions

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

In 4d,.
In 5d,.
In 6d,.
is the critical volume fraction, valid for overlapping objects.
For void models, is the critical void fraction, and is the total volume of the overlapping objects

Thresholds on hypercubic lattices

d	z	Site thresholds	Bond thresholds
4	8	0.198 0.197, 0.1968861, 0.196889, 0.196901, 0.19680, 0.1968904, 0.19688561	0.1600, 0.16005, 0.1601314, 0.160130, 0.1601310, 0.1601312, 0.16013122
5	10	0.141,0.198 0.141, 0.1407966, 0.1407966, 0.14079633	0.1181, 0.118, 0.11819, 0.118172, 0.1181718 0.11817145
6	12	0.106, 0.108, 0.109017, 0.1090117, 0.109016661	0.0943, 0.0942, 0.0942019, 0.09420165
7	14	0.05950, 0.088939, 0.0889511, 0.0889511, 0.088951121,	0.0787, 0.078685, 0.0786752, 0.078675230
8	16	0.0752101, 0.075210128	0.06770, 0.06770839, 0.0677084181
9	18	0.0652095, 0.0652095348	0.05950, 0.05949601, 0.0594960034
10	20	0.0575930, 0.0575929488	0.05309258, 0.0530925842
11	22	0.05158971, 0.0515896843	0.04794969, 0.04794968373
12	24	0.04673099, 0.0467309755	0.04372386, 0.04372385825
13	26	0.04271508, 0.04271507960	0.04018762, 0.04018761703

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

d	lattice	z	Site thresholds	Bond thresholds
4	diamond	5	0.2978	0.2715
4	kagome	8	0.2715	0.177
4	bcc	16	0.1037	0.074, 0.074212
4	fcc, D₄, hypercubic 2NN	24	0.0842, 0.08410, 0.0842001	0.049, 0.049517, 0.0495193
4	hypercubic NN+2NN	32	0.06190, 0.0617731	0.035827, 0.0338047
4	hypercubic 3NN	32	0.04540
4	hypercubic NN+3NN	40	0.04000	0.0271892
4	hypercubic 2NN+3NN	56	0.03310	0.0194075
4	hypercubic NN+2NN+3NN	64	0.03190, 0.0319407	0.0171036
4	hypercubic NN+2NN+3NN+4NN	88	0.0231538	0.0122088
4	hypercubic NN+...+5NN	136	0.0147918	0.0077389
4	hypercubic NN+...+6NN	232	0.0088400	0.0044656
4	hypercubic NN+...+7NN	296	0.0070006	0.0034812
4	hypercubic NN+...+8NN	320	0.0064681	0.0032143
4	hypercubic NN+...+9NN	424	0.0048301	0.0024117
5	diamond	6	0.2252	0.2084
5	kagome	10	0.2084	0.130
5	bcc	32	0.0446	0.033
5	fcc, D₅, hypercubic 2NN	40	0.0431, 0.0435913	0.026, 0.0271813
5	hypercubic NN+2NN	50	0.0334	0.0213
6	diamond	7	0.1799	0.1677
6	kagome	12	0.1677
6	fcc, D₆	60	0.0252, 0.02602674	0.01741556
6	bcc	64	0.0199
6	E₆	72	0.02194021	0.01443205
7	fcc, D₇	84	0.01716730	0.012217868
7	E₇	126	0.01162306	0.00808368
8	fcc, D₈	112	0.01215392	0.009081804
8	E₈	240	0.00576991	0.004202070
9	fcc, D₉	144	0.00905870	0.007028457
9		272	0.00480839	0.0037006865
10	fcc, D₁₀	180	0.007016353	0.005605579
11	fcc, D₁₁	220	0.005597592	0.004577155
12	fcc, D₁₂	264	0.004571339	0.003808960
13	fcc, D₁₃	312	0.003804565	0.0032197013

Thresholds in one-dimensional long-range percolation

In a one-dimensional chain we establish bonds between distinct sites and with probability decaying as a power-law with an exponent. Percolation occurs at a critical value for. The numerically determined percolation thresholds are given by:

		Critical thresholds as a function of. The dotted line is the rigorous lower bound.
0.1	0.047685
0.2	0.093211
0.3	0.140546
0.4	0.193471
0.5	0.25482
0.6	0.327098
0.7	0.413752
0.8	0.521001
0.9	0.66408

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.
Note: is the Schläfli symbol, signifying a hyperbolic lattice in which n regular m-gons meet at every vertex
For bond percolation on, we have by duality. For site percolation, because of the self-matching of triangulated lattices.
Cayley tree with coordination number

Thresholds for directed percolation

Lattice	z	Site percolation threshold	Bond percolation threshold
-d honeycomb	1.5	0.8399316, 0.839933, of -d sq.	0.8228569, 0.82285680
-d kagome	2	0.7369317, 0.73693182	0.6589689, 0.65896910
-d square, diagonal	2	0.705489, 0.705489, 0.70548522, 0.70548515, 0.7054852,	0.644701, 0.644701, 0.644701, 0.6447006, 0.64470015, 0.644700185, 0.6447001, 0.643
-d triangular	3	0.595646, 0.5956468, 0.5956470	0.478018, 0.478025, 0.4780250 0.479
-d simple cubic, diagonal planes	3	0.43531, 0.43531411	0.382223, 0.38222462 0.383
-d square nn	4	0.3445736, 0.344575 0.3445740	0.2873383, 0.287338 0.28733838 0.287
-d fcc			0.199)
-d hypercubic, diagonal	4	0.3025, 0.30339538	0.26835628, 0.2682
-d cubic, nn	6	0.2081040	0.1774970
-d bcc	8	0.160950, 0.16096128	0.13237417
-d hypercubic, diagonal	5	0.23104686	0.20791816, 0.2085
-d hypercubic, nn	8	0.1461593, 0.1461582	0.1288557
-d bcc	16	0.075582, 0.0755850, 0.07558515	0.063763395
-d hypercubic, diagonal	6	0.18651358	0.170615155, 0.1714
-d hypercubic, nn	10	0.1123373	0.1016796
-d hypercubic bcc	32	0.035967, 0.035972540	0.0314566318
-d hypercubic, diagonal	7	0.15654718	0.145089946, 0.1458
-d hypercubic, nn	12	0.0913087	0.0841997
-d hypercubic bcc	64	0.017333051	0.01565938296
-d hypercubic, diagonal	8	0.135004176	0.126387509, 0.1270
-d hypercubic,nn	14	0.07699336	0.07195
-d bcc	128	0.008 432 989	0.007 818 371 82

nn = nearest neighbors. For a -dimensional hypercubic system, the hypercube is in d dimensions and the time direction points to the 2D nearest neighbors.

Directed percolation with multiple neighbors

-d square with z NN, square lattice for z odd, tilted square lattice for z even

Lattice	z	Site percolation threshold	Bond percolation threshold
-d square	3		0.4395,
-d square	5		0.2249
-d square	7		0.1470
-d square	9		0.1081
-d square	11		0.0851
-d square	13		0.0701
-d tilted sq	2		0.6447
-d tilted sq	4		0.3272
-d tilted sq	6		0.2121
-d tilted sq	8		0.1553
-d tilted sq	10		0.1220
-d tilted sq	12		0.0999

For large z, p_c ~ 1/z

Site-Bond Directed Percolation

p_b = bond threshold
p_s = site threshold
Site-bond percolation is equivalent to having different probabilities of connections:
P₀ = probability that no sites are connected = + p_s²
P₂ = probability that exactly one descendant is connected to the upper vertex = p_s p_b
P₃ = probability that both descendants are connected to the original vertex = p_s p_b²
Normalization: P₀ + 2P₂ + P₃ = 1

Lattice	z	p_s	p_b	P₀	P₂	P₃
-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 and directed percolation. On an oriented system such as shown in the figure "d Square Lattice" above, we consider the down probability
p↓ = p p_d and the up probability p↑ = p, with p representing the average bond occupation probability and p_d controlling the anisotropy. When p_d = 0 or 1, we have pure DP, while when p_d = 1/2 we have the random diode model or essentially OP, with the threshold twice the OP value. For other values of p_d, we have a mixture of the two types of percolation. For a given p_d, the critical values of p = p_c are given below:

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

Exact critical manifolds of inhomogeneous systems

Inhomogeneous triangular lattice bond percolation
Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation
Inhomogeneous lattice, site percolation
or
Inhomogeneous union-jack lattice, site percolation with probabilities
Inhomogeneous martini lattice, bond percolation
Inhomogeneous martini lattice, site percolation. r = site in the star
Inhomogeneous martini-A lattice, bond percolation. Left side :. Right side:. Cross bond:.
Inhomogeneous martini-B lattice, bond percolation
Inhomogeneous martini lattice with outside enclosing triangle of bonds, probabilities from inside to outside, bond percolation
Inhomogeneous checkerboard lattice, bond percolation
Inhomogeneous bow-tie lattice, bond percolation
where are the four bonds around the square and is the diagonal bond connecting the vertex between bonds and.

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. Another way to say this is that constraints are sufficient to eliminate all zero-frequency vibrational modes, transforming a mechanically floppy network into one capable of supporting stress.
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:
2d
Bond threshold, triangular lattice: p_c = 0.6602 0.6602778
Site percolation, triangular lattice p_c = 0.69755, 0.6975
Correlation-length exponent: ν = 1.16, 1.19, 1.21, 1/ν = 0.850
? = -0.48
β = 0.175
Fractal dimension d_f = 1.86. 1.853, 1.850
Backbone fractal dimension d_b = 1.80, 1.78
Arbibi Sahimi 93: 2d bond tri: p„=0.641, site: p„=0.713.
Chubynsky and Thorpe 07. 3d: bond fcc, p_c = 0.495. bcc: p_c = 0.7485
Javerzam arXiv:2301.07614v2. 2d hull fractal dimension : d_f = 1.355
Roux, Hansen 88: central force elastic network: p* = 0.642, flv = 3.0, glv = 0.97 ;
Arababi, Sahimi 88:. 3d bond cubic elastic network p_c = 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.