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 pc, large clusters and long-range connectivity first appears, and this is called the percolation threshold. Depending on the method for obtaining the random network, one distinguishes between the site percolation threshold and the bond percolation threshold. More general systems have several probabilities p1, p2, etc., and the transition is characterized by a critical surface or manifold. One can also consider continuum systems, such as overlapping disks and spheres placed randomly, or the negative space.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.
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.
Simply duality in two dimensions implies that all fully triangulated lattices all have site thresholds of 1/2, and self-dual lattices have bond thresholds of 1/2.
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.
Percolation on 2D lattices
Thresholds on Archimedean lattices
This is a picture of the 11 Archimedean Lattices or uniform tilings, in which all polygons are regular and each vertex is surrounded by the same sequence of polygons. The notation for example means that every vertex is surrounded by four triangles and one hexagon. Drawings from. See also Uniform tilings.| Lattice | z | Site percolation threshold | Bond percolation threshold | |
| 3-12 or | 3 | 3 | 0.807900764... = 1/2 | 0.74042195, 0.74042077 0.740420800, 0.7404207988509, 0.740420798850811610, |
| cross, truncated trihexagonal | 3 | 3 | 0.7478008, 0.747806 | 0.6937314, 0.69373383, 0.693733124922 |
| square octagon, bathroom tile, 4-8, truncated square | 3 | - | 0.7297232, 0.729724 | 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 | 0.62181207, 0.621819 | 0.52483258, 0.5248311, 0.524831461573 |
| square | 4 | 4 | 0.59274, 0.59274605079210, 0.59274601, 0.59274605095, 0.59274621, 0.59274621, 0.59274598, 0.59274605, 0.593, 0.591, 0.569 | 1/2 |
| snub hexagonal, maple leaf | 5 | 5 | 0.579498 | 0.43430621, 0.43432764, 0.4343283172240, |
| snub square, puzzle | 5 | 5 | 0.550806 | 0.41413743, 0.4141378476, 0.4141378565917, |
| frieze, elongated triangular | 5 | 5 | 0.550213, 0.5502 | 0.4196, 0.41964191, 0.41964044, 0.41964035886369 |
| triangular | 6 | 6 | 1/2 | 0.347296355... = 2 sin, 1 + p3 − 3p = 0 |
Note: sometimes "hexagonal" is used in place of honeycomb, although in some fields, a triangular lattice is also called a hexagonal lattice. z = bulk coordination number.
2d lattices with extended and complex neighborhoods
In this section, sq corresponds to square-NN+2NN+3NN, etc.| Lattice | z | Site percolation threshold | Bond percolation threshold |
| sq,,,: square-NN, 2NN, 3NN, 5NN | 4 | 0.592... | |
| sq,,: NN+2NN, 2NN+3NN, 3NN+5NN | 8 | 0.407... | 0.25036834, 0.2503685 |
| sq: NN+3NN | 8 | 0.337 | 0.2214995 |
| sq: 2NN+5NN | 8 | 0.337 | |
| hc: honeycomb-NN+2NN+3NN | 12 | 0.300 | |
| tri: triangular-NN+2NN | 12 | 0.295, | |
| tri: triangular-2NN+3NN | 12 | 0.232020, | |
| sq: square-4NN | 8 | 0.270... | |
| sq: square-NN+5NN | 8 | 0.277 | |
| sq: square-NN+2NN+3NN | 12 | 0.292, 0.290 0.288, | 0.1522203 |
| sq: square-2NN+3NN+5NN | 12 | 0.288 | |
| sq: square-NN+4NN | 12 | 0.236 | |
| sq: square-2NN+4NN | 12 | 0.225 | |
| tri: triangular-4NN | 12 | 0.192450 | |
| tri: triangular-NN+2NN+3NN | 18 | 0.225, 0.215459 | |
| square: 3NN+4NN | 12 | 0.221 | |
| square: NN+2NN+5NN | 12 | 0.240 | 0.13805374 |
| square: NN+3NN+5NN | 12 | 0.233 | |
| square: 4NN+5NN | 12 | 0.199 | |
| square: NN+2NN+4NN | 16 | 0.219 | |
| square: NN+3NN+4NN | 16 | 0.208 | |
| square: 2NN+3NN+4NN | 16 | 0.202 | |
| square: NN+4NN+5NN | 16 | 0.187 | |
| square: 2NN+4NN+5NN | 16 | 0.182 | |
| square: 3NN+4NN+5NN | 16 | 0.179 | |
| square: NN+2NN+3NN+5NN | 16 | 0.208 | 0.1032177 |
| triangular: 4NN+5NN | 18 | 0.140250, | |
| square: NN+2NN+3NN+4NN | 20 | 0.196 0.196724 | 0.0841509 |
| square: NN+2NN+4NN+5NN | 20 | 0.177 | |
| square: NN+3NN+4NN+5NN | 20 | 0.172 | |
| square: 2NN+3NN+4NN+5NN | 20 | 0.167 | |
| square: NN+2NN+3NN+5NN+6NN | 20 | 0.0783110 | |
| square: NN+2NN+3NN+4NN+5NN | 24 | 0.164 | |
| triangular: NN+4NN+5NN | 24 | 0.131660 | |
| square: NN+...+6NN | 28 | 0.0558493 | |
| triangular: 2NN+3NN+4NN+5NN | 30 | 0.117460 | |
| triangular: NN+2NN+3NN+4NN+5NN | 36 | 0.115740 | |
| square: NN+...+7NN | 36 | 0.04169608 | |
| square: square distance ≤ 4 | 40 | 0.105 | |
| square: NN+..+8NN | 44 | 0.095765 | |
| square: NN+..+9NN | 48 | 0.02974268 | |
| square: NN+...+11NN | 60 | 0.0230119 | |
| square: NN+...+32NN | 224 | 0.0053050415 708 15 1.102 812 | |
| square: NN+...+86NN | 708 | 0.001557644 | |
| square: NN+...+141NN | 1224 | 0.000880188 | |
| square: NN+...+185NN | 1652 | 0.000645458 | |
| square: NN+...+317NN | 3000 | 0.000349601 | |
| square: NN+...+413NN | 4016 | 0.0002594722 | |
| square: square distance ≤ 6 | 84 | 0.049 | |
| square: square distance ≤ 8 | 144 | 0.028 | |
| square: square distance ≤ 10 | 220 | 0.019 | |
| 2x2 overlapping squares* | 0.58365 | ||
| 3x3 overlapping squares* | 0.59586 |
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 squares, given here is the net fraction of sites occupied similar to the in continuum percolation. The case of a 2×2 system is equivalent to percolation of a square lattice: NN+2NN+3NN+4NN with threshold with. The 3×3 system corresponds to sq with z=44 and. For larger overlapping squares, see.
Approximate formulas for thresholds of Archimedean lattices
Site-bond percolation in 2D
Site bond percolation.| 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 |
Approximate formula for a honeycomb lattice
| Lattice | z | Threshold | Notes | |
| honeycomb | 3 | 3 | when equal: ps = pb = 0.82199 | approximate formula, ps = site prob., pb = bond prob., t = 1 − 2 sin |
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 | 3⅓ | 0.6501834, 0.650184 | 0.585863... = 1 − pcbond |
| Pentagonal D=+ | 3,4 | 3⅓ | 0.6470471, 0.647084, 0.6471 | 0.580358... = 1 − pcbond, 0.5800 |
| D=+ | 3,6 | 3 3/5 | 0.639447 | 0.565694... = 1 − pcbond |
| dice, rhombille tiling D = + | 3,6 | 4 | 0.5851, 0.585040 | 0.475595... = 1 − pcbond |
| ruby dual D = + + | 3,4,6 | 4 | 0.582410 | 0.475167... = 1 − pcbond |
| union jack, tetrakis square tiling D = + | 4,8 | 6 | 1/2 | 0.323197... = 1 − pcbond |
| bisected hexagon, cross dual D= ++ | 4,6,12 | 6 | 1/2 | 0.306266... = 1 − pcbond |
| asanoha D=+ | 3,12 | 6 | 1/2 | 0.259579... = 1 − pcbond |
2-uniform lattices
Top 3 lattices: #13 #12 #36Bottom 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 1/7 | 0.672929..., 1 − 2p3 − 2p4 − 2p5 − 7p6 + 18p7 + 11p8 − 35p9 + 21p10 − 4p11 = 0 | |
| bow-tie | 3,4 | 3⅓ | 0.625457..., 1 − 2p2 − 3p3 + 4p4 − p5 = 0 | |
| martini-A + | 3,4 | 3⅓ | 1/ | 0.625457..., 1 − 2p2 − 3p3 + 4p4 − p5 = 0 |
| bow-tie dual | 3,4 | 3⅔ | 0.595482..., 1-pcbond | |
| bow-tie | 3,4,6 | 3⅔ | 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 | 1/2 |
| bow-tie dual | 3,4,8 | 4 2/5 | 0.466787..., 1 − pcbond | |
| 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 − pcbond | |
| bow-tie dual | 3,6,10 | 5½ | 0.547... = pcsite | 0.327071..., 1 − pcbond |
| martini dual + | 3,9 | 6 | 1/2 | 0.292893... = 1 − 1/ |
Thresholds on 2D covering, medial, and matching lattices
| Lattice | z | Site percolation threshold | Bond percolation threshold | |
| covering/medial | 4 | 4 | pcbond = 0.693731... | 0.5593140, 0.559315 |
| covering/medial, square kagome | 4 | 4 | pcbond = 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 | 1/2 | 0.3371 |
| square matching lattice | 8 | 8 | 1 − pcsite = 0.407253... | 0.25036834 |
covering/medial lattice
covering/medial lattice
covering/medial lattice, equivalent to the kagome subnet, and in black, the dual of these lattices.
covering/medial lattice, medial dual, shown in red, with medial lattice in light gray behind it. The pattern on the left appears in Iranian tilework.
Thresholds on 2D chimera non-planar lattices
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 | pcbond = 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 | pcbond = 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, 0.5157 |
| linear 4-mers on a triangular lattice | 6 | 0.5220 |
| linear 8-mers on a triangular lattice | 6 | 0.5281 |
| linear 12-mers on a triangular lattice | 6 | 0.5298 |
| linear 16-mers on a triangular lattice | 6 | 0.5328 |
| linear 32-mers on a triangular lattice | 6 | 0.5407 |
| linear 64-mers on a triangular lattice | 6 | 0.5455 |
| linear 80-mers on a triangular lattice | 6 | 0.5500 |
| 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 dimers 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 on 2D inhomogeneous lattices
Thresholds for 2D continuum models
| System | Φc | ηc | nc |
| Disks of radius r | 0.67634831, 0.6763475, 0.676339, 0.6764, 0.6766, 0.676, 0.679, 0.674 0.676, | 1.12808737, 1.128085, 1.128059, 1.13, 0.8 | 1.43632545, 1.436322, 1.436289, 1.436320, 1.436323, 1.438, 1.216 |
| Ellipses, ε = 1.5 | 0.0043 | 0.00431 | 2.059081 |
| Ellipses, ε = 5/3 | 0.65 | 1.05 | 2.28 |
| Ellipses, aspect ratio ε = 2 | 0.6287945, 0.63 | 0.991000, 0.99 | 2.523560, 2.5 |
| Ellipses, ε = 3 | 0.56 | 0.82 | 3.157339, 3.14 |
| Ellipses, ε = 4 | 0.5 | 0.69 | 3.569706, 3.5 |
| Ellipses, ε = 5 | 0.455, 0.455, 0.46 | 0.607 | 3.861262, 3.86 |
| Ellipses, ε = 10 | 0.301, 0.303, 0.30 | 0.358 0.36 | 4.590416 4.56, 4.5 |
| Ellipses, ε = 20 | 0.178, 0.17 | 0.196 | 5.062313, 4.99 |
| Ellipses, ε = 50 | 0.081 | 0.084 | 5.393863, 5.38 |
| Ellipses, ε = 100 | 0.0417 | 0.0426 | 5.513464, 5.42 |
| Ellipses, ε = 200 | 0.021 | 0.0212 | 5.40 |
| Ellipses, ε = 1000 | 0.0043 | 0.00431 | 5.624756, 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 | 0.66675, 0.66674349, 0.66653, 0.6666, 0.668 | 1.09884280, 1.0982, 1.098 | 1.09884280, 1.0982, 1.098 |
| Randomly oriented squares | 0.62554075, 0.6254 0.625, | 0.9822723, 0.9819 0.982278 | 0.9822723, 0.9819 0.982278 |
| Rectangles, ε = 1.1 | 0.624870 | 0.980484 | 1.078532 |
| Rectangles, ε = 2 | 0.590635 | 0.893147 | 1.786294 |
| Rectangles, ε = 3 | 0.5405983 | 0.777830 | 2.333491 |
| Rectangles, ε = 4 | 0.4948145 | 0.682830 | 2.731318 |
| Rectangles, ε = 5 | 0.4551398, 0.451 | 0.607226 | 3.036130 |
| Rectangles, ε = 10 | 0.3233507, 0.319 | 0.3906022 | 3.906022 |
| Rectangles, ε = 20 | 0.2048518 | 0.2292268 | 4.584535 |
| Rectangles, ε = 50 | 0.09785513 | 0.1029802 | 5.149008 |
| Rectangles, ε = 100 | 0.0523676 | 0.0537886 | 5.378856 |
| Rectangles, ε = 200 | 0.02714526 | 0.02752050 | 5.504099 |
| Rectangles, ε = 1000 | 0.00559424 | 0.00560995 | 5.609947 |
| Sticks of length | 5.6372858, 5.63726, 5.63724 | ||
| Power-law disks, x=2.05 | 0.993 | 4.90 | 0.0380 |
| Power-law disks, x=2.25 | 0.8591 | 1.959 | 0.06930 |
| Power-law disks, x = 2.5 | 0.7836 | 1.5307 | 0.09745 |
| Power-law disks, x = 4 | 0.69543 | 1.18853 | 0.18916 |
| Power-law disks, x = 5 | 0.68643 | 1.1597 | 0.22149 |
| Power-law disks, x = 6 | 0.68241 | 1.1470 | 0.24340 |
| Power-law disks, x=7 | 0.6803 | 1.140 | 0.25933 |
| Power-law disks, x=8 | 0.67917 | 1.1368 | 0.27140 |
| Power-law disks, x = 9 | 0.67856 | 1.1349 | 0.28098 |
| Voids around disks of radius r | 1 − Φc = 0.32355169, 0.318, 0.3261 |
equals critical total area for disks, where N is the number of objects and L is the system size.
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.
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
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, |
Thresholds on 3D lattices
| Lattice | z | filling factor* | filling fraction* | Site percolation threshold | Bond percolation threshold | - | |
| -a oxide | 23 32 | 2.4 | 0.748713 | = 1/2 = 0.742334 | - | ||
| -b oxide | 23 32 | 2.4 | 0.233 | 0.174 | 0.745317 | = 1/2 = 0.739388 | - |
| silicon dioxide | 4,22 | 2 ⅔ | 0.638683 | - | |||
| Modified -b | 32,2 | 2 ⅔ | 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, 0.426, 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 2/3 | 0.3904 | 0.2350 | - | |||
| 3D kagome | 6 | π / 12 = 0.37024 | 0.1442 | 0.3895 =pc for diamond dual and pc for diamond lattice | 0.2709 | - | |
| Bow-tie stack dual | 5⅓ | 0.3480 | 0.2853 | - | |||
| honeycomb stack | 5 | 5 | 0.3701 | 0.3093 | - | ||
| octagonal stack dual | 5 | 5 | 0.3840 | 0.3168 | - | ||
| pentagonal stack | 5⅓ | 0.3394 | 0.2793 | - | |||
| kagome stack | 6 | 6 | 0.453450 | 0.1517 | 0.3346 | 0.2563 | - |
| fcc dual | 42,8 | 5 1/3 | 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 1/3 | 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 | 12 | 12 | π / = 0.740480 | 0.147530 | 0.195, 0.198, 0.1998, 0.1992365, 0.19923517, 0.1994 | 0.1198 0.1201635 | - |
| hcp | 12 | 12 | π / = 0.740480 | 0.147545 | 0.195, 0.1992555 | 0.1201640 0.119 | - |
| La2−x Srx Cu O4 | 12 | 12 | 0.19927 | - | |||
| simple cubic with 2NN | 12 | 12 | 0.1991 | - | |||
| simple cubic with NN+4NN | 12 | 12 | 0.15040 | 0.1068263 | - | ||
| simple cubic with 3NN+4NN | 14 | 14 | 0.20490 | 0.1012133 | - | ||
| bcc NN+2NN | 14 | 14 | 0.175, 0.1686 | 0.0991 | - | ||
| 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 | 0.0752326 | - | ||
| fcc with NN+2NN | 18 | 18 | 0.136 | - | |||
| 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 | 0.0497080 | - | ||
| bcc with NN+2NN+3NN | 26 | 26 | 0.095 | - | |||
| simple cubic with NN+2NN+3NN+4NN | 32 | 32 | 0.10000 | 0.0392312 | - | ||
| fcc with NN+2NN+3NN | 42 | 42 | 0.061, 0.0610 | ||||
| fcc with NN+2NN+3NN+4NN | 54 | 54 | 0.0500 |
Filling factor = fraction of space filled by touching spheres at every lattice site. Also called Atomic Packing Factor.
Filling fraction = filling factor * pc.
NN = nearest neighbor, 2NN = next-nearest neighbor, 3NN = next-next-nearest neighbor, etc.
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 Φc |
| percolating excluded volume of athermal polymer matrix | 0.4304 |
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 | Φc | ηc |
| Spheres of radius r | 0.2895, 0.2896, 0.289573, 0.2896, 0.2854 | 0.3418, 0.341889, 0.3360, 0.34189, |
| Oblate ellipsoids with major radius r and aspect ratio 4/3 | 0.2831 | 0.3328 |
| Prolate ellipsoids with minor radius r and aspect ratio 3/2 | 0.2757, 0.2795 | 0.3278 |
| Oblate ellipsoids with major radius r and aspect ratio 2 | 0.2537, 0.2629 | 0.3050 |
| Prolate ellipsoids with minor radius r and aspect ratio 2 | 0.2537, 0.2618, 0.25 | 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 |
| 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 | |
| Voids around disks of radius r | 22.86 | |
| Voids around oblate ellipsoids of major radius r and aspect ratio 10 | 15.42 | |
| Voids around oblate ellipsoids of major radius r and aspect ratio 2 | 6.478 | |
| Voids around hemispheres | 0.0455 | |
| Voids around aligned tetrahedra | 0.0605 | |
| Voids around rotated tetrahedra | 0.0605 | |
| Voids around aligned cubes | 0.036, 0.0381 | |
| Voids around rotated cubes | 0.0381 | |
| Voids around aligned octahedra | 0.0407 | |
| Voids around rotated octahedra | 0.0398 | |
| Voids around aligned dodecahedra | 0.0356 | |
| Voids around rotated dodecahedra | 0.0360 | |
| Voids around aligned icosahedra | 0.0346 | |
| Voids around rotated icosahedra | 0.0336 | |
| Voids around spheres | 0.034, 0.032, 0.030, 0.0301, 0.0294, 0.0300, 0.0317, 0.0308 0.0301 | 3.506, 3.515 |
| Jammed spheres | 0.183, 0.1990, see also contact network of jammed spheres | 0.59 |
is the total volume, where N is the number of objects and L is the system size.
is the critical volume fraction.
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.
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 3D correlated percolation
| Lattice | z | Site percolation threshold | Bond percolation threshold | |
| Drilling percolation, simple cubic lattice | 6 | 6 | *0.633965, 0.6339 , 6345 |
- In drilling percolation, p is the fraction of columns that have not been removed
Thresholds in different dimensional spaces
Continuum models in higher dimensions
| d | System | Φc | ηc |
| 4 | Overlapping hyperspheres | 0.1223 | 0.1304 |
| 4 | Aligned hypercubes | 0.1132, 0.1132348 | 0.1201 |
| 4 | Voids around hyperspheres | 0.00211 | 6.161 |
| 5 | Overlapping hyperspheres | 0.05443 | |
| 5 | Aligned hypercubes | 0.04900, 0.0481621, | 0.05024 |
| 5 | Voids around hyperspheres | 1.26x10−4 | 8.98 |
| 6 | Overlapping hyperspheres | 0.02339 | |
| 6 | Aligned hypercubes | 0.02082, 0.0213479 | 0.02104 |
| 6 | Voids around hyperspheres | 8.0x10−6 | 11.74 |
| 7 | Overlapping hyperspheres | 0.02339 | |
| 7 | Aligned hypercubes | 0.00999, 0.0097754 | 0.01004 |
| 8 | Overlapping hyperspheres | 0.004904 | |
| 8 | Aligned hypercubes | 0.004498 | |
| 9 | Overlapping hyperspheres | 0.002353 | |
| 9 | Aligned hypercubes | 0.002166 | |
| 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.
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 | 0.16005, 0.1601314, 0.160130, 0.1601310,, 0.1601312, 0.16013122 |
| 5 | 10 | 0.141,0.198 0.141, 0.1407966, 0.1407966, 0.14079633 | 0.11819, 0.118172, 0.1181718 0.11817145 |
| 6 | 12 | 0.106, 0.108, 0.109017, 0.1090117, 0.109016661 | 0.0942, 0.0942019, 0.09420165 |
| 7 | 14 | 0.05950, 0.088939, 0.0889511, 0.0889511, 0.088951121, | 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.
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 | 24 | 0.0842, 0.08410 | 0.049, 0.049517 |
| 4 | cubic NN+2NN | 32 | 0.06190 | 0.035827 |
| 4 | cubic 3NN | 32 | 0.04540 | |
| 4 | cubic NN+3NN | 40 | 0.04000 | |
| 4 | cubic 2NN+3NN | 58 | 0.03310 | |
| 4 | cubic NN+2NN+3NN | 64 | 0.03190 | |
| 5 | diamond | 6 | 0.2252 | 0.2084 |
| 5 | kagome | 10 | 0.2084 | 0.130 |
| 5 | bcc | 32 | 0.0446 | 0.033 |
| 5 | fcc | 40 | 0.0431 | 0.026 |
| 6 | diamond | 7 | 0.1799 | 0.1677 |
| 6 | kagome | 12 | 0.1677 | |
| 6 | fcc | 60 | 0.0252 | |
| 6 | bcc | 64 | 0.0199 |
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:| 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 z: pc = 1 /
Cayley tree with a distribution of z with mean, mean-square pc=
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.
Exact critical manifolds of inhomogeneous systems
Inhomogeneous triangular lattice bond percolationInhomogeneous 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.