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Percolation threshold

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Percolation threshold is a mathematical term related to percolation theory, which is the formation of long-range connectivity in random systems. In engineering and coffee making, percolation is the slow flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems, 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.

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[edit] 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, long-range connectivity first appears, and this is called the 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).

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. [1] 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.

The purpose of this page is to gather in one place the most up-to-date precise values of percolation thresholds and critical surfaces, including all the exact results that are known.

The notation such as (4,82) comes from Grünbaum and Shepard, [2] 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.

[edit] Thresholds on 2d regular and Archimedean lattices


This is a picture of the 11 Archimedean Lattices, in which all polygons are regular and each vertex is surrounded by the same sequence of polygons. The notation (34 , 6) for example means that every vertex is surrounded by four triangles and one hexagon.


Lattice z Site Percolation Threshold Bond Percolation Threshold
(3, 122 ) 3 0.807900764... = (1 - 2 sin (π/18))1/2 [3] 0.74042195(80)[4],

0.74042081(10) [5]
(4, 6, 12) 3 0.747806(4) [3] 0.69373383(72)[4]
(4, 82) 3 0.729724(3) [3] 0.67680232(63)[4]
honeycomb (63) 3 0.697043(3)[3] 0.6970413(10) [5] 0.652703645... = 1-2 sin (π/18), 3p2 − p3 = 1 [6]
kagomé (3, 6, 3, 6) 4 0.652703645... = 1 - 2 sin(π/18) [6] 0.5244053(3) [7],

0.52440516(10) [5], 0.52440499(2)[8]
(3, 4, 6, 4) 4 0.621819(3) [3] 0.52483258(53)[4]
square (44) 4 0.59274621(13) [9], 0.59274621(33) [10], 0.59274598(4) [11][12], 0.59274605(3)[8] 1/2
(34,6 ) 5 0.579498(3) [3] 0.43430621(50)[4]
(32, 4, 3, 4 ) 5 0.550806(3) [3] 0.41413743(46)[4]
(33, 42) 5 0.550213(3) [3] 0.41964191(43) [4]
triangular (36) 6 1/2 0.347296355... = 2 sin (π/18), 3p − p3 = 1 [6]

Note: sometimes "hexagonal" is used in place of honeycomb, although in some fields, a triangular lattice is also called "hexagonal" (as in hexagonal lattice). z = bulk coordination number.

Also Im[(16 + 16*I)^(2/9)] = 2 Im[(-1)^(1/18)] = 0.3472963553338

[edit] 2-Uniform Lattices

Top 3 Lattices: (1/2)(36)+(1/2)(34,6) (1/4)(36)+(3/4)(34,6) (1/7)(36)+(6/7)(32,4,12)
Bottom 3 Lattices: (1/7)(36)+(6/7)(32,62) (1/2)(33)+(1/2)(42;3,4,6,4) (1/2)(34,6)+(1/2)(32,62)

20 2 uniform lattices

[2]
Top 2 Lattices: (2/3)(3,42,6)+(1/3)(3,4,6,4) (1/2)(32,4,3,4)+(1/2)(3,4,6,4)
Bottom 2 Lattices: (1/2)(3,4,3,12)+(1/2)(3,122) (1/3)(3,4,6,4)+(2/3)(4,6,12)

20 2 uniform lattices

[2]
Top 4 Lattices: (2/3)(33,42)+(1/3)(44) (1/2)(33,42)+(1/2)(44) (1/3)(36)+(2/3)(33,42) (1/2)(36)+(1/2)(33,42)
Bottom 3 Lattices: (4/5)(3,42,6)+(1/5)(3,6,3,6) (4/5)(3,42,6)+(1/5)(3,6,3,6)* (2/3)(32,62)+(1/3)(3,6,3,6)

20 2 uniform lattices

[2]
Top 2 Lattices: (1/7)(36)+(6/7)(32,4,3,4) (1/3)(33,42)+(2/3)(32,4,3,4)
Bottom Lattice: (1/2)(33,42)+(1/2) (32,4,3,4)

20 2 uniform lattices

[2]

# Lattice z, z Site Percolation Threshold Bond Percolation Threshold
41 (1/2)(3,4,3,12) + (1/2)(3, 122) 4,3 0.7680(2)[13] 0.67493(5) [14]
42 (1/3)(3,4,6,4) + (2/3)(4,6,12) 4,3 0.7157(2) [13] 0.63909(5) [14]
36 (1/7)(36) + (6/7)(32,4,12) 6,4 0.6808(2) [13] 0.55778(5) [14]
15 (2/3)(32,62) + (1/3)(3,6,3,6) 4,4 0.6499(2) [13] 0.53632487(40) [14]
34 (1/7)(36) + (6/7)(32,6) 6,3 0.6329(2) [13] 0.51708(5) [14]
16 (4/5)(3,42,6) + (1/5)(3,6,3,6) 4,4 0.6286(2) [13] 0.51891529(35) [14]
17 (4/5)(3,42,6) + (1/5)(3,6,3,6)* 4,4 0.6279(2) [13] 0.51769(5) [14]
35 (2/3)(3,42,6) + (1/3)(3,4,6,4) 4,4 0.6221(2) [13] 0.51973831(40) [14]
11 (1/2)(34,6) + (1/2)(32,62) 5,4 0.6171(2) [13] 0.48921(5) [14]
37 (1/2)(33,42) + (1/2)(3,4,6,4) 5,4 0.5885(2) [13] 0.47351(5) [14]
30 (1/2)(32,4,3,4) + (1/2)(3,4,6,4) 5,4 0.5883(2) [13] 0.46573(5) [14]
23 (1/2)(33,42) + (1/2)(44) 5,4 0.5720(2) [13] 0.45845(5) [14]
22 (2/3)(33,42) + (1/3)(44) 5,4 0.5648(2) [13] 0.44529(5) [14]
12 (1/4)(36) + (3/4)(34,6) 6,5 0.5607(2) [13] 0.41110(5) [14]
33 (1/2)(33,42) + (1/2)(32,4,3,4) 5,5 0.5505(2) [13] 0.41628(5) [14]
32 (1/3)(33,42) + (2/3)(32,4,3,4) 5,5 0.5504(2) [13] 0.41549(5) [14]
31 (1/7)(36) + (6/7)(32,4,3,4) 6,5 0.5440(2) [13] 0.40380(5) [14]
13 (1/2)(36) + (1/2)(34,6) 6,5 0.5407(2) [13] 0.38915(5) [14]
21 (1/3)(36) + (2/3)(33,42) 6,5 0.5342(2) [13] 0.39492(5) [14]
20 (1/2)(36) + (1/2)(33,42) 6,5 0.5258(2) [13] 0.38285(5) [14]

[edit] Thresholds on 2d bowtie and martini lattices

To the left, center, and right are: the martini lattice, the martini-A lattice, the martini-B lattice.

Example image caption


Lattice z Site Percolation Threshold Bond Percolation Threshold
martini 3 0.764826..., 3p3 - p4 = 1 [15] 0.707107... = 1/\sqrt{2} [16]
bowtie 5 0.5472(2) [17] 0.404518..., p + 6 p2 - 6 p3 + p5 = 1 [18]
martini-A 3, 4 1/\sqrt{2} = 0.707107... [15] [19] 0.625457..., p5 -4p4+3p3+2p2 = 1 [16] [19]
martini-B 3, 5 1/(1 +\sqrt{5})/2..., p2+p = 1 [15] [19] 1/2 [16] [19]

[edit] Thresholds on other 2d lattices

Lattice z Site Percolation Threshold Bond Percolation Threshold
square covering lattice 6 1/2 0.3371(1) [20]
dice (kagome dual) (1/3)(4^6)+(2/3)(4^3) (1/3)6+(2/3)3=4 0.5851(4) [21] 0.47550501(2) [8]

[edit] Thresholds on subnet lattices

Example image caption


Example image caption


Lattice z Site Percolation Threshold Bond Percolation Threshold
checkerboard - 2 x 2 subnet 4,3 0.596303(1) [22]
checkerboard - 4 x 4 subnet 4,3 0.633685(9) [22]
checkerboard - 8 x 8 subnet 4,3 0.642318(5) [22]
checkerboard - 16 x 16 subnet 4,3 0.64237(1) [22]
checkerboard- 32 x 32 subnet 4,3 0.64219(2) [22]
checkerboard - \infty subnet 4,3 0.642216(10) [22]
kagome - 2 x 2 subnet 6,4 0.6008624(10) [5]
kagome - 3 x 3 subnet 6,4 0.6193296(10) [5]
kagome - 4 x 4 subnet 6,4 0.625365(3) [5]
kagome - \infty subnet 6,4 0.628961(2) [5]
triangular - 2 x 2 subnet 6,4 0.471628788 [22]
triangular - 3 x 3 subnet 6,4 0.509077793 [22]
triangular - 4 x 4 subnet 6,4 0.524364822 [22]
triangular - 5 x 5 subnet 6,4 0.5315976(10) [22]
triangular - \infty subnet 6,4 0.53993(1) [22]

[edit] 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. [23]

l (polymer length) z Bond Percolation
1 4 0.5(exact) [24]
2 4 0.47697(4)[24]
4 4 0.44892(6) [24]
8 4 0.41880(4)[24]


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

k z Site Thresholds Bond Thresholds
1 4 0.593(2) [25] 0.5009(2) [25]
2 4 0.564(2) [25] 0.4859(2) [25]
3 4 0.552(2) [25] 0.4732(2) [25]
4 4 0.542(2) [25] 0.4630(2) [25]
5 4 0.531(2) [25] 0.4565(2) [25]
6 4 0.522(2) [25] 0.4497(2) [25]
7 4 0.511(2) [25] 0.4423(2) [25]
8 4 0.502(2) [25] 0.4348(2) [25]
9 4 0.493(2) [25] 0.4291(2) [25]
10 4 0.488(2) [25] 0.4232(2) [25]
11 4 0.482(2) [25] 0.4159(2) [25]
12 4 0.476(2) [25] 0.4114(2) [25]
13 4 0.471(2) [25] 0.4061(2) [25]
14 4 0.467(2) [25] 0.4011(2) [25]
15 4 0.4011(2) [25] 0.3979(2) [25]

[edit] Thresholds on 2d inhomogeneous lattices

Lattice z Site Percolation Threshold Bond Percolation Threshold
bowtie with p = 1/2 on one non-diagonal bond 3 0.3819654(5) [26]

[edit] Thresholds for 2d continuum models

System Φc ηc
Disks of radius r 0.6763475(6) [27]
Voids around disks of radius r 0.159(2) [28]

ηc = πr2N / L2 equals critical total area, where N is the number of objects and L is the system size.

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

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

File:Voronoi.pdf

[edit] Thresholds on 2d random and quasi-lattices

Lattice z Site Percolation Threshold Bond Percolation Threshold
Voronoi tessellation 3 0.71410(2)[29] 0.666931(2) [29] [30]
Voronoi covering 4 0.666931(2)[29] [30] 0.53618(2) [29]
Penrose rhomb dual 4 0.6381(3)[21] 0.5233(2) [21]
Penrose rhomb 4 (average) 0.5837(3)[21], 0.58391(1)[31] 0.4770(2) [21]
Delaunay triangulation 6 (average) 1/2 0.333069(2) [29][30]

[edit] Thresholds on 3d lattices

Lattice z Site Percolation Threshold Bond Percolation Threshold Dimer Percolation Threshold
ice 4 0.433(11)[32] 0.388(10)[33]
diamond 4 0.426(+0.08,-0.02) [34]0.4301(4)[35] 0.390(11) [33],0.3893(2)[35]
simple cubic 6 0.311604(6)[36], 0.311605(5)[37], 0.311600(5)[38],

0.3116081(21)[39], 0.3116080(4)[40], 0.3116004(35)[41]

 0.2488126(5) [42] 0.2555(1)[43]
Icosahedral Penrose 6 (average) 0.285[44] 0.225 [44]
Penrose w/2 diagonals 6.764 (average) 0.271[44] 0.207 [44]
bcc 8 0.2459615(10)[40] 0.1802875(10)[42]
fcc 12 0.1992365(10)[40] 0.1201635(10)[42]
hcp 12 0.1992555(10)[45] 0.1201640(10)[45]
La2-x Srx Cu O4 12 0.19927(2) [46]
Penrose w/8 diagonals 12.764 (average) 0.188[44] 0.111 [44]
cubic with n.n.n. 18 0.13735(5) [47]
cubic with n.n.n.n. 26 0.0976445(10) [47]

n.n.n. = next-nearest neighbor, n.n.n.n. = next-next nearest neighbor

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?

[edit] Thresholds for 3d continuum models

System Φc ηc
Spheres of radius r 0.289573(2) [48] 0.341889(3) [48]
Voids around spheres of radius r 0.030(2) [28], 0.0301(3) [49], 0.0294 [50], 0.0298(5) [51] 3.514 (16) [51]
Randomly oriented disks of radius r 0.9614(5)[52]
Randomly oriented square plates of side \sqrt{\pi} r 0.8647(6)[52]
Randomly oriented triangular plates of side \sqrt{2 \pi} /3^{1/4} r 0.7295(6)[52]

ηc = (4 / 3)πr3N / L3 is the total volume, where N is the number of objects and L is the system size.

\phi_c = 1 - e^{-\eta_c} is the critical volume fraction.

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.

[edit] Thresholds on hypercubic lattices

d z Site Thresholds Bond Thresholds
4 8 0.1968861(14) [53], 0.196889(3) [54] 0.1601314(13) [53], 0.160130(3) [54]
5 10 0.1407966(15) [53] 0.118172(1) [53]
6 12 0.109017(2) [53] 0.0942019(6) [53]
7 14 0.0889511(9) [53]
0.088939(20) [55]		 0.0786752(3) [53]
8 16 0.0752101(5) [53] 0.06770839(7) [53]
9 18 0.0652095(3) [53] 0.05949601(5) [53]
10 20 0.0575930(1) [53] 0.05309258(4) [53]
11 22 0.05158971(8) [53] 0.04794969(1) [53]
12 24 0.04673099(6) [53] 0.04372386(1) [53]
13 26 0.04271508(8) [53] 0.04018762(1) [53]
d z Site Thresholds Bond Thresholds τ
4 8 0.196889(3) [54] 0.160130(3) [54] 2.313(3) [54]
5 10 0.14081(1) [54] 0.118174(4) [54] 2.412(4) [54]

Simulation parameters and results for pc and the Fisher exponent τ.

d z Site Thresholds Bond Thresholds zspread dmin
4 8 0.196889 [54] 0.160130 [54] 0.622(2) [54] 1.607(5) [54]
5 10 0.14081 [54] 0.118174 [54] 0.552(2) [54] 1.812(6) [54]

Simulation parameters and results for the spreading exponent zspread and shortest path exponent.

[edit] Thresholds on kagomé lattices in higher dimensions

d z Site Thresholds Bond Thresholds rw
3 6 0.3895(2) [56] 0.417(1) [56]
4 8 0.2715(3) [56] 0.274(1) [56]
5 10 0.2084(4) [56] 0.208(1) [56]
6 12 0.1677(7) [56] 0.170(1) [56]

[edit] Thresholds for directed percolation

Lattice z Site Percolation Threshold Bond Percolation Threshold
(1+1)-d square lattice 2 0.70548522(4) [57] 0.64470015(5) [58]
(1+1)-d triangular lattice 3 0.5956468(5) [58] 0.478025(1) [58]
(2+1)-d bcc lattice 4 0.2873383(1) [59], 0.287338(3) [60]
(3+1)-d simple hypercubic lattice 4 0.3025(10) [61]
(4+1)-d simple hypercubic lattice 8 0.1461582(3) [62]
(4+1)-d body-centered hypercubic lattice 16 0.0755850(3) [62]


[edit] Thresholds for contact process

Lattice z λc
1-d 2 3.29785(2) [63]

Inactive site becomes active at rate λn/z where n is the number of active nearest neighbors; active sites become inactive at unit rate.

[edit] General formulas for exact results

Inhomogeneous triangular lattice bond percolation[6]

1 − p1 − p2 − p3 + p1p2p3 = 0

Inhomogeneous honeycomb lattice bond percolation = kagomé lattice site percolation[6]

1 − p1p2 − p2p3 − p1p3 + p1p2p3 = 0

Inhomogeneous martini lattice (bond percolation) [64]

1 − (p1p2r3 + p2p3r1 + p1p3r2) − (p1p2r1r2 + p1p3r1r3 + p2p3r2r3) + p1p2p3(r1r2 + r1r3 + r2r3) + r1r2r3(p1p2 + p1p3 + p2p3) − 2p1p2p3r1r2r3 = 0

Inhomogeneous martini lattice (site percolation). r = site in the star

1 − r(p1p2 + p1p3 + p2p3 − p1p2p3) = 0

Inhomogeneous martini-A (3-7) lattice

Inhomogeneous martini-B (3-5) lattice

Inhomogeneous checkerboard lattice (conjecture) [65]

1 − (p1p2 + p1p3 + p1p4 + p2p3 + p2p4 + p3p4) + p1p2p3 + p1p2p4 + p1p3p4 + p2p3p4 = 0

[edit] See also

[edit] References

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  2. ^ a b c d e Grünbaum, Branko; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1. 
  3. ^ a b c d e f g h Suding, P. N.; R. M. Ziff (1999). "Site percolation thresholds for Archimedean lattices". Physical Review E 60 (1): 275–283. doi:10.1103/PhysRevE.60.275. 
  4. ^ a b c d e f g Parviainen, Robert (2007). "Estimation of bond percolation thresholds on the Archimedean lattices". J. Phys. A 40: 9253--9258. doi:10.1088/1751-8113/40/31/005. 
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  6. ^ a b c d e Sykes, M. F.; J. W. Essam (1964). "Exact critical percolation probabilities for site and bond problems in two dimensions". Journal of Mathematical Physics (N.Y.) 5 (8): 1117–1127. doi:10.1063/1.1704215. 
  7. ^ Ziff, R. M.; P. W. Suding (1997). "Determination of the bond percolation threshold for the kagomé lattice". Journal of Physics A 30: 5351–5359. doi:10.1088/0305-4470/30/15/021. 
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  10. ^ de Oliveira, P.M.C.; R. A. Nobrega, D. Stauffer. (2003). "Corrections to finite size scaling in percolation". Brazilian Journal of Physics 33: 616–618. doi:10.1590/S0103-97332003000300025. 
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  15. ^ a b c Scullard, C. R. (2006). "Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation". Physical Review E 73: 016107. doi:10.1103/PhysRevE.73.016107. 
  16. ^ a b c Ziff, R. M. (2006). "Generalized cell–dual-cell transformation and exact thresholds for percolation". Physical Review E 73: 016134. doi:10.1103/PhysRevE.73.016134. 
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  18. ^ Wierman, J. C. (1984). "A bond percolation critical probability determination based on the star-triangle transformation". Journal of Physics A 17: 1525–1530. doi:10.1088/0305-4470/17/7/020. 
  19. ^ a b c d Ziff, R. M.; Scullard, C. R. (2006). "Exact bond percolation thresholds in two dimensions". Journal of Physics A 39 (49): 15087. doi:10.1088/0305-4470/39/49/003. 
  20. ^ Ziff, R. M. (2007). To be published. 
  21. ^ a b c d e Sakamoto, S.; F. Yonezawa and M. Hori (1989). "A proposal for the estimation of percolation thresholds in two-dimensional lattices". J. Phys. A 22 (14): L699-L704. doi:10.1088/0305-4470/22/14/009. 
  22. ^ a b c d e f g h i j k Haji Akbari, Amir; R. M. Ziff (2008). To be published. 
  23. ^ Zia, R. K. P.; W. Yong, B. Schmittmann (2009). "Percolation of a collection of finite random walks: a model for gas permeation through thin polymeric membranes". Journal of Mathematical Chemistry 45: 58–64. doi:10.1007/s10910-008-9367-6. 
  24. ^ a b c d Wu, Yong; B. Schmittmann, R. K. P. Zia (2008). "Two-dimensional polymer networks near percolation". Journal of Physics A 41 (2): 025008. doi:10.1088/1751-8113/41/2/025004. 
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