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. Below the threshold a giant connected component does not exist while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability p, or more generally a critical surface for a group of parameters p1, p2, ..., such that infinite connectivity (percolation) first occurs.

Contents

[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, large clusters and 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 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 Archimedean lattices

Example image caption


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 (34, 6) for example means that every vertex is surrounded by four triangles and one hexagon. Drawings from .[3] See also Uniform Tilings.


    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    (3, 122 ) 3 3 0.807900764... = (1 - 2 sin (π/18))1/2 [4] 0.74042195(80),[5]

    0.74042081(10),[6] 0.74042077(2),[7]
    (4, 6, 12) 3 3 0.747806(4) [4] 0.69373383(72)[5]
    (4, 82) 3 3 0.729724(3) [4] 0.67680232(63)[5]
    honeycomb (63) 3 3 0.697043(3)[4] 0.6970413(10) [6] 0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=0[8]
    kagomé (3, 6, 3, 6) 4 4 0.652703645... = 1 - 2 sin(π/18) [8] 0.524404978(5),[7] 0.52440499(2),[9]

    0.52440516(10),[6] 0.5244053(3) [10]
    (3, 4, 6, 4) 4 4 0.621819(3) [4] 0.52483258(53)[5]
    square (44) 4 4 0.59274621(13),[11] 0.59274621(33),[12] 0.59274598(4),[13][14] 0.59274605(3)[9] 1/2
    maple leaf [15] (34,6 ) 5 5 0.579498(3) [4] 0.43430621(50)[5]
    snub square, puzzle (32, 4, 3, 4 ) 5 5 0.550806(3) [4] 0.41413743(46)[5]
    (33, 42) 5 5 0.550213(3) [4] 0.41964191(43) [5]
    triangular (36) 6 6 1/2 0.347296355... = 2 sin (π/18), 1+ p3-3p=0[8]

    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.

    [edit] Square lattice with complex neighborhoods

    Lattice z Site Percolation Threshold Bond Percolation Threshold
    square: 3N, 4N, 6N 4 0.592...[16][17]
    square: 3N+2N, 4N+3N, 6N+4N 8 0.407...[16][17]
    square: 4N+2N 8 0.337...[16][17]
    square: 6N+3N 8 0.337...[17]
    square: 5N 8 0.270...[17]
    square: 6N+2N 8 0.277...[17]
    square: 4N+3N+2N 12 0.288...[16][17]
    square: 6N+4N+3N 12 0.288...[17]
    square: 5N+2N 12 0.236...[17]
    square: 5N+3N 12 0.225...[17]
    square: 5N+4N 12 0.221...[17]
    square: 6N+3N+2N 12 0.240...[17]
    square: 6N+4N+2N 12 0.233...[17]
    square: 6N+5N 12 0.199...[17]
    square: 5N+3N+2N 16 0.219...[17]
    square: 5N+4N+2N 16 0.208...[17]
    square: 5N+4N+3N 16 0.202...[17]
    square: 6N+5N+2N 16 0.187...[17]
    square: 6N+5N+3N 16 0.182...[17]
    square: 6N+5N+4N 16 0.179...[17]
    square: 6N+4N+3N+2N 16 0.208...[17]
    square: 5N+4N+3N+2N 20 0.196...[17]
    square: 6N+5N+3N+2N 20 0.177...[17]
    square: 6N+5N+4N+2N 20 0.172...[17]
    square: 6N+5N+4N+3N 20 0.167...[17]
    square: 6N+5N+4N+3N+2N 24 0.164...[17]

    2N = nearest neighbours, 3N = next-nearest neighbours, 4N = next-next-nearest neighbours, 5N = next-next-next-nearest neighbours, etc.

    [edit] Approximate formulas for thresholds of Archimedean lattices (Under construction)

    Lattice z Site Percolation Threshold Bond Percolation Threshold
    (3, 122 ) 3
    (4, 6, 12) 3
    (4, 82) 3 0.676835..., 4p3 + 3p4 - 6 p5- 2 p6 = 1 [18]
    honeycomb (63) 3
    kagomé (3, 6, 3, 6) 4 0.524430..., 3p2 + 6p3 - 12 p4+ 6 p5 - p6 = 1 [19]
    (3, 4, 6, 4) 4
    square (44) 4 1/2 (exact)
    (34,6 ) 5 0.434371..., 12p3 + 36p4 -21 p5- 327 p6 + 69p7 + 2532p8 - 6533 p9

    + 8256 p10 - 6255p11 + 2951p12 - 837 p13+ 126 p14 - 7p15= 1 [20]
    snub square, puzzle (32, 4, 3, 4 ) 5
    (33, 42) 5
    triangular (36) 6 1/2 (exact)

    Archimedean Duals (Laves Lattices)

    Example image caption

    Laves lattices are the duals to the Archimedean lattices. Drawings from.[3] See also Uniform Tilings.

    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    Cairo pentagonal

    D(32,4,3,4)=(2/3)(53)+(1/3)(54)

    		 3,4 3⅓ 0.650184[3] pcbond=1-pcbond(32,4,3,4)

    0.58586256(54)[5]
    D(33,42)=(1/3)(54)+(2/3)(53) 3,4 3⅓ 0.647084[3] pcbond=1-Pcbond(33,42)

    0.58035808(57)[5]
    D(34,6)=(1/5)(46)+(4/5)(43) 3,6 3 3/5 0.639447[3] pcbond=1-pcbond(34,6 )

    0.56569378(50)[5]
    dice

    D(3,6,3,6)=(1/3)(46)+(2/3)(43)

    		 3,6 4 0.5851(4),[21] 0.585040[3] pcbond=1-pcbond(3,6,3,6 )

    0.475595021(5),[7] 0.47559500(8),[9] 0.47559483(90),[6] 0.475594(7)[10]
    ruby[22]

    D(3,4,6,4)=(1/6)(46)+(2/6)(43)+(3/6)(44)

    		 3,4,6 4 0.582410[3] Pcbond=1-Pcbond(3,4,6,4 )

    0.47516741(47)[5]
    cross

    D(4,6,12)= (1/6)(312)+(2/6)(36)+(1/2)(34)

    		 4,6,12 6 1/2 pcbond=1-pcbond(4,6,12)

    0.30626616(28)[5]
    asanoha [23]

    D(3, 122)=(2/3)(33)+(1/3)(312)

    		 3,12 6 1/2 pcbond=1-pcbond(3, 122)

    =0.25957804(20),[5] 0.25957918(90),[6] 0.25957922(8)[7]
    union jack

    D(4,82 )=(1/2)(34)+(1/2)(38)

    		 4,8 6 1/2 pcbond=1-pcbond(4,82 )

    0.23219767(37)[5]

    Site bond percolation (both thresholds apply simultaneously to one system).

    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    square 4 4 0.615185(15)[24] 0.95
    0.667280(15)[24] 0.85
    0.732100(15)[24] 0.75
    0.75 0.726195(15)[24]
    0.815560(15)[24] 0.65
    0.85 0.615810(30)[24]
    0.95 0.533620(15)[24]

    [edit] 2-Uniform Lattices

    Top 3 Lattices: #13 #12 #36
    Bottom 3 Lattices: #34 #37 #11

    20 2 uniform lattices

    [2]
    Top 2 Lattices: #35 #30
    Bottom 2 Lattices: #41 #42

    20 2 uniform lattices

    [2]
    Top 4 Lattices: #22 #23 #21 #20
    Bottom 3 Lattices: #16 #17 #15

    20 2 uniform lattices

    [2]
    Top 2 Lattices: #31 #32
    Bottom Lattice: #33

    20 2 uniform lattices

    [2]

    # Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    41 (1/2)(3,4,3,12) + (1/2)(3, 122) 4,3 0.7680(2)[25] 0.67493252(36) [26]
    42 (1/3)(3,4,6,4) + (2/3)(4,6,12) 4,3 0.7157(2) [25] 0.64536587(40) [26]
    36 (1/7)(36) + (6/7)(32,4,12) 6,4 0.6808(2) [25] 0.55778329(40) [26]
    15 (2/3)(32,62) + (1/3)(3,6,3,6) 4,4 0.6499(2) [25] 0.53632487(40) [26]
    34 (1/7)(36) + (6/7)(32,62) 6,3 0.6329(2) [25] 0.51707873(70) [26]
    16 (4/5)(3,42,6) + (1/5)(3,6,3,6) 4,4 0.6286(2) [25] 0.51891529(35) [26]
    17 (4/5)(3,42,6) + (1/5)(3,6,3,6)* 4,4 0.6279(2) [25] 0.51769462(35) [26]
    35 (2/3)(3,42,6) + (1/3)(3,4,6,4) 4,4 0.6221(2) [25] 0.51973831(40) [26]
    11 (1/2)(34,6) + (1/2)(32,62) 5,4 0.6171(2) [25] 0.48921280(37) [26]
    37 (1/2)(33,42) + (1/2)(3,4,6,4) 5,4 0.5885(2) [25] 0.47229486(38) [26]
    30 (1/2)(32,4,3,4) + (1/2)(3,4,6,4) 5,4 0.5883(2) [25] 0.46573078(72) [26]
    23 (1/2)(33,42) + (1/2)(44) 5,4 0.5720(2) [25] 0.45844622(40) [26]
    22 (2/3)(33,42) + (1/3)(44) 5,4 0.5648(2) [25] 0.44528611(40) [26]
    12 (1/4)(36) + (3/4)(34,6) 6,5 0.5607(2) [25] 0.41109890(37) [26]
    33 (1/2)(33,42) + (1/2)(32,4,3,4) 5,5 0.5505(2) [25] 0.41628021(35) [26]
    32 (1/3)(33,42) + (2/3)(32,4,3,4) 5,5 0.5504(2) [25] 0.41549285(36) [26]
    31 (1/7)(36) + (6/7)(32,4,3,4) 6,5 0.5440(2) [25] 0.40379585(40) [26]
    13 (1/2)(36) + (1/2)(34,6) 6,5 0.5407(2) [25] 0.38914898(35) [26]
    21 (1/3)(36) + (2/3)(33,42) 6,5 0.5342(2) [25] 0.39491996(40) [26]
    20 (1/2)(36) + (1/2)(33,42) 6,5 0.5258(2) [25] 0.38285085(38) [26]

    [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. Below: the martini covering lattice, same as the 2x2, 1x1 subnet for kagome-type lattices.

    Example image caption


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

    Example image caption
    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    martini (3/4)(3,92)+(1/4)(93) 3 3 0.764826..., 1 +p4 - 3p3=0[27] 0.707107... = 1/√2 [28]
    bow-tie (c) 3,4 3 1/7 0.672929..., 1-2p3-2p4-2p5-7p6+18p7+11p8-35p9+21p10-4p11=0 [29]
    bow-tie (d) 3,4 3⅓ 0.625457..., 1-2p2-3p3+4p4-p5=0 [29]
    martini-A (2/3)(3,72)+(1/3)(3,73) 3,4 3⅓ 1/√2[29] 0.625457..., 1-2p2-3p3+4p4-p5=0 [29]
    bow-tie dual lattice (e) 3,4 3⅔ 0.595482..., 1-pcbond (bow-tie (a)) [29]
    bow-tie (b) 3,4,6 3⅔ 0.533213..., 1-p- 2p3 -4p4-4p5+156+ 13p7-36p8+19p9+ p10 + p11=0 [29]
    martini covering (1/2)(33,9)+(1/2)(3,9,3,9) 4 4 0.707107... = 1/√2 [28] 0.57086651(33) [30]
    martini-B (1/2)(3,5,3,52)+(1/2)(3,52) 3, 5 4 0.618034... = 2/(1 +√5)..., 1- p2-p=0[27][29] 1/2 [28][29]
    bow-tie dual lattice (f) 3,4,8 4 2/5 0.466787..., 1-pcbond (bow-tie (b))[29]
    bow-tie (a) (1/2)(32,4,32,4)+(1/2)(3,4,3) 4,6 5 0.5472(2) [31] 0.404518..., 1 - p - 6p2 +6p3-p5=0 [29]
    bow-tie dual lattice (h) 3,6,8 5 0.374543..., 1-pcbond(bow-tie (d))[29]
    bow-tie dual lattice (g) 3,6,10 5½ 0.327071..., 1-pcbond(bow-tie (c))[29]

    [edit] Thresholds on other 2d lattices

    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    square covering lattice (non-planar) 6 6 1/2 0.3371(1) [32]
    square matching lattice (non-planar) 8 8 0.40725395(3) [9] 0.25036834(6) [9]

    [edit] Thresholds on subnet lattices

    Example image caption
    Example image caption

    The 2 × 2 subnet is known as the "triangular kagome" lattice [33]

    Example image caption
    Lattice z Site Percolation Threshold Bond Percolation Threshold
    checkerboard – 2 × 2 subnet 4,3 0.596303(1) [34]
    checkerboard – 4 × 4 subnet 4,3 0.633685(9) [34]
    checkerboard – 8 × 8 subnet 4,3 0.642318(5) [34]
    checkerboard – 16 × 16 subnet 4,3 0.64237(1) [34]
    checkerboard- 32 × 32 subnet 4,3 0.64219(2) [34]
    checkerboard – \infty subnet 4,3 0.642216(10) [34]
    kagome – 2 × 2 subnet 4 Same as bond perc. on (3, 122) 0.6008624(10),[6] 0.60086193(3) [7]
    kagome – 3 × 3 subnet 4 0.6193296(10),[6] 0.61933176(5),[7] 0.61933044(32)[35]
    kagome – 4 × 4 subnet 4 0.625365(3),[6] 0.62536424(7)[7]
    kagome – \infty subnet 4 0.628961(2) [6]
    kagome – (1 × 1):(3 × 3) subnet 4,3 0.728355596425196...[7] 0.58609776(37) [35]
    kagome – (1 × 1):(4 × 4) subnet 0.738348473943256...[7]
    kagome – (1 × 1):(5 × 5) subnet 0.743548682503071...[7]
    kagome – (1 × 1):(6 × 6) subnet 0.746418147634282...[7]
    kagome – (2 × 2):(3 × 3) subnet 0.61091770(30) [35]
    triangular – 2 × 2 subnet 6,4 0.471628788 [34]
    triangular – 3 × 3 subnet 6,4 0.509077793 [34]
    triangular – 4 × 4 subnet 6,4 0.524364822 [34]
    triangular – 5 × 5 subnet 6,4 0.5315976(10) [34]
    triangular – \infty subnet 6,4 0.53993(1) [34]

    [edit] Thresholds of dimers a square lattice

    system z Site Threshold
    unoriented dimers 4 0.5617 [36]
    parallel dimers 4 0.5683[36]

    [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. [37]

    l (polymer length) z Bond Percolation
    1 4 0.5(exact) [38]
    2 4 0.47697(4)[38]
    4 4 0.44892(6) [38]
    8 4 0.41880(4)[38]

    [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) [39] 0.5009(2) [39]
    2 4 0.564(2) [39] 0.4859(2) [39]
    3 4 0.552(2) [39] 0.4732(2) [39]
    4 4 0.542(2) [39] 0.4630(2) [39]
    5 4 0.531(2) [39] 0.4565(2) [39]
    6 4 0.522(2) [39] 0.4497(2) [39]
    7 4 0.511(2) [39] 0.4423(2) [39]
    8 4 0.502(2) [39] 0.4348(2) [39]
    9 4 0.493(2) [39] 0.4291(2) [39]
    10 4 0.488(2) [39] 0.4232(2) [39]
    11 4 0.482(2) [39] 0.4159(2) [39]
    12 4 0.476(2) [39] 0.4114(2) [39]
    13 4 0.471(2) [39] 0.4061(2) [39]
    14 4 0.467(2) [39] 0.4011(2) [39]
    15 4 0.4011(2) [39] 0.3979(2) [39]

    [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) [40]

    [edit] Thresholds for 2d continuum models

    System Φc ηc nc
    Aligned squares of side \ell=2a 0.6666(4) [41] 1.098(1) [41]
    Randomly oriented squares 0.6254(2) [41] 0.9819(6) [41]
    Disks of radius r 0.6763475(6) [42] 1.128085(2) 1.466322(2)
    Ellipses of aspect ratio 2 0.63 [43] 0.76 1.94
    Ellipses of aspect ratio 5 0.455 [44] 0.607 3.864
    Ellipses of aspect ratio 10 0.301 [44] 0.358 4.56
    Ellipses of aspect ratio 20 0.178 [44] 0.196 4.99
    Ellipses of aspect ratio 50 0.081 [44] 0.084 5.38
    Ellipses of aspect ratio 100 0.0417 [44] 0.0426 5.42
    Ellipses of aspect ratio 1000 0.0043 [44] 0.00431 5.5
    Sticks of length \ell 5.63726(6) [45]
    Voids around disks of radius r 0.159(2) [46]

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

    ηc = πabN / L2 for ellipses of semi-major and semi-minor axes of a and b, respectively. Aspect ratio ε = a / b

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

    n_c = \ell^2 N / L^2 equals number of objects of length \ell = 2 a per unit area.

    For ellipses, nc = (4ε / π)ηc

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

    For more ellipse values, see [43]

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

    Left to right: (a) Voronoi diagram (solid lines) and the dual Delaunay triangulation (dotted lines) for a Poisson distribution of points, (b) Delaunay triangulation only, (c) Voronoi diagram (black lines) and the covering or line graph (dotted red lines).

    Example image caption

    Voronoi.png VoronoiCov12.png

    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    Voronoi tessellation 3 3 0.71410(2)[47], 0.7151* [25] 0.68,[48] 0.666931(5),[47] 0.6670(1) [49]
    Voronoi covering 4 4 0.666931(2)[47][49] 0.53618(2) [47]
    Penrose rhomb dual 4 4 0.6381(3)[21] 0.5233(2) [21]
    Penrose rhomb 4 4 0.5837(3),[21] 0.58391(1)[50] 0.4770(2) [21]
    Delaunay triangulation 6 6 1/2 [51] 0.333069(2) [47][49]

    *Theoretical estimate

    [edit] Thresholds on slabs

    Lattice z Site Percolation Threshold Bond Percolation Threshold
    h= 2, SC, open b.c. 0.47424 [52]
    h = 3, BCC, periodic b.c. 0.21113018(38) [53]
    h = 4, BCC, periodic b.c. 0.20235168(59) [53]
    h= 4, SC, open b.c. 0.3997 [52]
    h = 5, SC, periodic b.c. 0.278102(5) [53]
    h = 6, SC, periodic b.c. 0.272380(2) [53]
    h = 7, SC, periodic b.c. 5,6 0.3459514(12) [53] 0.268459(1) [53]
    h= 8, SC, open b.c. .0.3557 [52]
    h = 8, SC, periodic b.c. 0.265615(5) [53]

    More for SC open b.c. in Ref.[52]

    h is the thickness of the slab, h x ∞ x ∞.

    [edit] Thresholds on 3d lattices

    Lattice z Site Percolation Threshold Bond Percolation Threshold Dimer Percolation Threshold
    ice 4 0.433(11)[54] 0.388(10)[55]
    diamond 4 0.426(+0.08,-0.02) [56] 0.4301(4)[57] 0.390(11),[55] 0.3893(2)[57]
    simple cubic 6 0.311604(6),[58] 0.311605(5),[59] 0.311600(5),[60]

    0.3116077(4),[61] 0.3116081(13),[62] 0.3116080(4),[63] 0.3116004(35)[64]

    		 0.2488126(5) [65] 0.2488125(25) [66] 0.2555(1)[67]
    Icosahedral Penrose 6 (average) 0.285[68] 0.225 [68]
    Penrose w/2 diagonals 6.764 (average) 0.271[68] 0.207 [68]
    bcc 8 0.2459615(10),[63] 0.2460(3),[69] 0.2464(7) [70] 0.1802875(10)[65]
    fcc 12 0.1992365(10)[63] 0.1201635(10)[65]
    hcp 12 0.1992555(10)[71] 0.1201640(10)[71]
    La2-x Srx Cu O4 12 0.19927(2) [72]
    Penrose w/8 diagonals 12.764 (average) 0.188[68] 0.111 [68]
    simple cubic with short-length correlation 6+ 0.126(1)[73]
    simple cubic with 2NN 12 0.1991(1) [74]
    simple cubic with 3NN 8 0.2455(1) [74]
    simple cubic with NN+2NN 18 0.1372(1),[74] 0.13735(5) [75]
    simple cubic with NN+3NN 14 0.1420(1) [74]
    simple cubic with 2NN+3NN 20 0.1036(1) [74]
    simple cubic with NN+2NN+3NN 26 0.0976(1),[74] 0.0976445(10) [75]

    NN = nearest neighbor, 2NN = next-nearest neighbor, 3NN = 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
    Aligned cubes of side \ell = 2 a 0.2773(2) [41] 0.3248(3)[41]
    Randomly oriented cubes of side \ell = 2 a 0.2168(2) [41] 0.2444(3)[41]
    Spheres of radius r 0.289573(2) [76] 0.341889(3) [76]
    Voids around spheres of radius r (showing void fraction) 0.030(2),[46] 0.0301(3),[77] 0.0294,[78] 0.0300(3) [79] 3.506(8) [79]
    Randomly oriented disks of radius r (in 3D) 0.9614(5)[80]
    Randomly oriented square plates of side \sqrt{\pi} r 0.8647(6)[80]
    Randomly oriented triangular plates of side \sqrt{2 \pi} /3^{1/4} r 0.7295(6)[80]

    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),[81] 0.196889(3),[82] 0.196901(5) [83] 0.1601314(13),[81] 0.160130(3),[82] 0.1601310(10) [66]
    5 10 0.1407966(15) [81] 0.118172(1),[81] 0.1181718(3) [66]
    6 12 0.109017(2) [81] 0.0942019(6) [81]
    7 14 0.0889511(9),[81] 0.088939(20) [84] 0.0786752(3) [81]
    8 16 0.0752101(5) [81] 0.06770839(7) [81]
    9 18 0.0652095(3) [81] 0.05949601(5) [81]
    10 20 0.0575930(1) [81] 0.05309258(4) [81]
    11 22 0.05158971(8) [81] 0.04794969(1) [81]
    12 24 0.04673099(6) [81] 0.04372386(1) [81]
    13 26 0.04271508(8) [81] 0.04018762(1) [81]
    d z Site Thresholds Bond Thresholds τ
    4 8 0.196889(3) [82] 0.160130(3) [82] 2.313(3) [82]
    5 10 0.14081(1) [82] 0.118174(4) [82] 2.412(4) [82]

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

    d z Site Thresholds Bond Thresholds zspread dmin
    4 8 0.196889 [82] 0.160130 [82] 0.622(2) [82] 1.607(5) [82]
    5 10 0.14081 [82] 0.118174 [82] 0.552(2) [82] 1.812(6) [82]

    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) [85] 0.417(1) [85]
    4 8 0.2715(3) [85] 0.274(1) [85]
    5 10 0.2084(4) [85] 0.208(1) [85]
    6 12 0.1677(7) [85] 0.170(1) [85]

    [edit] Thresholds on hyperbolic, hierarchical, and tree lattices

    Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk [86]

    Example image caption


    Depiction of the non-planar Hanoi network HN-NP [87]

    Example image caption


    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    Lower Upper
    {4,5} hyperbolic 5 5 0.27[88] 0.52[88]
    {7,3} hyperbolic 3 3 0.72[88] 0.53[88]
    {3,7} hyperbolic 7 7 0.20[88] 0.37[88]
    {∞,3} Cayley tree 3 3 1/2[88] 1[88]
    Enhanced binary tree (EBT) 0.304(1)[88] 0.48,[88] 0.564(1)[89]
    Enhanced binary tree dual 0.436(1)[89] 0.696(1)[89]
    Non-Planar Hanoi Network (HN-NP) 0.319445[87] 0.381996[87]
    Cayley tree with grandparents 8 0.158656326[90]

    Note: {m,n} is the Shläfli symbol, signifying a hyperbolic lattice in which n regular m-gons meet at every vertex

    [edit] Thresholds for directed percolation

    Lattice z Site Percolation Threshold Bond Percolation Threshold
    (1+1)-d square, diagonal direction 2 0.705489(4),[91] 0.70548522(4) [92] 0.644701(2),[93] 0.644701(1),[94] 0.64470015(5),[95] 0.644700185(5)[92]
    (1+1)-d triangular 3 0.5956468(5) [95] 0.478025(1) [95]
    (2+1)-d simple cubic, diagonal planes 3 0.43531(1) [96] 0.382223(7) [96]
    (2+1)-d square nn (= bcc) 4 0.3445736(3),[97] 0.344575(15) [98] 0.2873383(1),[99] 0.287338(3) [96]
    (3+1)-d hypercubic, diagonal planes 4 0.3025(10) [100]
    (3+1)-d cubic, nn 6 0.2081040(4) [97] 0.1774970(5) [66]
    (3+1)-d body-centered hypercubic 8 0.160950(30) [98]
    (4+1)-d hypercubic, nn 8 0.1461593(2),[97] 0.1461582(3) [101] 0.1288557(5) [66]
    (4+1)-d body-centered hypercubic 16 0.075582(17) [98]

    0.0755850(3) [101]
    (5+1)-d hypercubic, nn 10 0.1123373(2) [97] 0.1016796(5) [66]
    (5+1)-d body-centered hypercubic 32 0.035967(23) [98]
    (6+1)-d hypercubic, nn 12 0.0913087(2) [97] 0.0841997(14) [66]
    (7+1)-d hypercubic,nn 14 0.07699336(7) [97] 0.07195(5) [66]

    nn = nearest neighbors. For a (d+1)-dimensional hypercubic system, the hypercube is in d dimensions and the time direction points to the 2d nearest neighbors.

    [edit] General formulas for exact results

    Inhomogeneous triangular lattice bond percolation[8]

    1 − p1 − p2 − p3 + p1p2p3 = 0

    Inhomogeneous honeycomb lattice bond percolation = kagomé lattice site percolation[8]

    1 − p1p2 − p2p3 − p1p3 + p1p2p3 = 0

    Inhomogeneous (3,12^2) lattice,[4] [102]

    1 − 3(s1s2)2 + (s1s2)3 = 0, or s1s2 = 1 − 2sin(π / 18)

    Inhomogeneous martini lattice (bond percolation) [29]

    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. Left side (top of "A" to bottom): r_2,\ p_1. Right side: r_1, \ p_2. Cross bond: \ r_3.

    1 − p1r2 − p2r1 − p1p2r3 − p1r1r3 − p2r2r3 + p1p2r1r3 + p1p2r2r3 + p1r1r2r3 + p2r1r2r3 − p1p2r1r2r3 = 0

    Inhomogeneous martini-B (3–5) lattice

    Inhomogeneous checkerboard lattice (conjecture) [19]

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

    [edit] Percolation thresholds of graphs

    For random graphs not embedded in space the percolation threshold can be calculated exactly. For example for random regular graphs where all nodes have the same degree k, pc=1/k. For Erdos - Reyni (ER) graphs with Poissonian degree distribution, pc=1/<k>.[103] The critical threshold was calculated exactly also for interdependent ER networks.[104]

    [edit] See also

    [edit] References

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