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.

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. Depending on the method for obtaining the random network, one distinguishes between the site percolation threshold and the bond percolation threshold. More general systems have several probabilities p1, p2, etc., and the transition is characterized by a critical surface or manifold. One can also consider continuum systems, such as overlapping disks and spheres placed randomly, or the negative space (Swiss-cheese models).

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.

Error bars in the last digit or digits are shown by numbers in parentheses. Thus, 0.729724(3) signifies 0.729724 ± 0.000003, and 0.74042195(80) signifies 0.74042195 ± 0.00000080. The error bars variously represent one or two standard deviations in net error (including statistical and expected systematic error), or an empirical confidence interval.

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 (<VAR >3</VAR >4, 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
    3-12 or (3, 122 ) 3 3 0.807900764... = (1 - 2 sin (π/18))1/2[4] 0.7404207988474(7),[5][6] 0.740420800(2),[7] 0.74042195(80),[8]

    0.74042077(2)[9]

    cross (4, 6, 12) 3 3 0.7478008(2),[5] 0.747806(4)[4] 0.6937314(1),[5] 0.69373383(72)[8]
    square octagon, bathroom tile, 4-8, truncated square

    (4, 82)

    		 3 3 0.7297233(5),[5] 0.729724(3)[4] 0.676803124(1),[5] 0.67680232(63)[8]
    honeycomb (63) 3 3 0.6962(6),[10] 0.6970402218(6),[5] 0.6970402(1),[11] 0.6970413(10),[12] 0.697043(3),[4] 0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=0[13]
    kagome (3, 6, 3, 6) 4 4 0.652703645... = 1 - 2 sin(π/18)[13] 0.524404978(5),[9] 0.52440499(2),[11]

    0.52440572...,[14] 0.52440500(1),[7] 0.52440516(10),[12] 0.5244053(3),[15] 0.524404999134(2) [5][6]

    ruby[16] (3, 4, 6, 4) 4 4 0.62181207(7),[5] 0.621819(3)[4] 0.5248311(1),[5] 0.52483258(53)[8]
    square (44) 4 4 0.592746010(2),[5] 0.59274621(13),[17] 0.59274621(33),[18] 0.59274598(4),[19][20] 0.59274605(3)[11] 1/2
    snub hexagonal, maple leaf [21] (34,6 ) 5 5 0.579498(3)[4] 0.43432764(3),[5] 0.43430621(50)[8]
    snub square, puzzle (32, 4, 3, 4 ) 5 5 0.550806(3)[4] 0.4141378476 (7),[5] 0.41413743(46)[8]
    (33, 42) 5 5 0.550213(3)[4] 0.41964044(1),[5] 0.41964191(43)[8]
    triangular (36) 6 6 1/2 0.347296355... = 2 sin (π/18), 1+ p3-3p=0[13]

    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.

    Square lattice with complex neighborhoods

    Lattice z Site Percolation Threshold Bond Percolation Threshold
    square: 3N, 4N, 6N 4 0.592...[22][23]
    square: 3N+2N, 4N+3N, 6N+4N 8 0.407...[22][23][24]
    square: 4N+2N 8 0.337...[22][23]
    square: 6N+3N 8 0.337...[23]
    square: 5N 8 0.270...[23]
    square: 6N+2N 8 0.277...[23]
    square: 4N+3N+2N 12 0.288...[22][23]
    square: 6N+4N+3N 12 0.288...[23]
    square: 5N+2N 12 0.236...[23]
    square: 5N+3N 12 0.225...[23]
    square: 5N+4N 12 0.221...[23]
    square: 6N+3N+2N 12 0.240...[23]
    square: 6N+4N+2N 12 0.233...[23]
    square: 6N+5N 12 0.199...[23]
    square: 5N+3N+2N 16 0.219...[23]
    square: 5N+4N+2N 16 0.208...[23]
    square: 5N+4N+3N 16 0.202...[23]
    square: 6N+5N+2N 16 0.187...[23]
    square: 6N+5N+3N 16 0.182...[23]
    square: 6N+5N+4N 16 0.179...[23]
    square: 6N+4N+3N+2N 16 0.208...[23]
    square: 5N+4N+3N+2N 20 0.196...[23]
    square: 6N+5N+3N+2N 20 0.177...[23]
    square: 6N+5N+4N+2N 20 0.172...[23]
    square: 6N+5N+4N+3N 20 0.167...[23]
    square: 6N+5N+4N+3N+2N 24 0.164...[23]

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

    Approximate formulas for thresholds of Archimedean lattices

    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 [25]
    honeycomb (63) 3
    kagome (3, 6, 3, 6) 4 0.524430..., 3p2 + 6p3 - 12 p4+ 6 p5 - p6 = 1 [26]
    (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 [27]

    snub square, puzzle (32, 4, 3, 4 ) 5
    (33, 42) 5
    triangular (36) 6 1/2 (exact)

    Formulas for site-bond percolation

    Lattice z \overline z Threshold Notes
    (63) honeycomb 3 3 bs[1-({\sqrt {t}}/(3-t))({\sqrt {b}}-{\sqrt {t}})]=t,

    when equal: b = s = 0.82199

    		 approximate formula, s = site prob., b = bond prob., t = 1 - 2 sin (π/18) [12]

    Archimedean Duals (Laves Lattices)

    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 0.6501834(2),[5] 0.650184[3] pcbond=1-pcbond(32,4,3,4)

    0.58586256(54)[8]

    D(33,42)=(1/3)(54)+(2/3)(53) 3,4 3⅓ 0.6470474(6),[28] 0.647084[3] pcbond=1-Pcbond(33,42)

    0.58035808(57)[8]

    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)[8]

    dice, rhombille tiling

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

    		 3,6 4 0.5851(4),[29] 0.585040[3] pcbond=1-pcbond(3,6,3,6 )

    0.475595021(5),[9] 0.47559500(8),[11] 0.47559483(90),[12] 0.475594(7)[15]

    ruby dual

    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)[8]

    bisected hexagon,[30] cross dual

    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)[8]

    asanoha (hemp leaf)[31]

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

    		 3,12 6 1/2 pcbond=1-pcbond(3, 122) = 0.25957804(20),[8] 0.25957918(90),[12] 0.25957922(8)[9]
    union jack, tetrakis square tiling

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

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

    0.23219767(37)[8]

    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)[32] 0.95
    0.667280(15)[32] 0.85
    0.732100(15)[32] 0.75
    0.75 0.726195(15)[32]
    0.815560(15)[32] 0.65
    0.85 0.615810(30)[32]
    0.95 0.533620(15)[32]

    * For more values, see An Investigation of site-bond percolation

    2-Uniform Lattices

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

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

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

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

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

    Inhomogeneous 2-Uniform Lattice

    This figure shows the 2-uniform lattice #37 in the isoradial representation in which each polygon is inscribed in a circle of unit radius. The squares in the 2-uniform lattice must now be represented as rectangles 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 (1/2)(33,42) + (1/2)(3,4,6,4), while the dual lattice has vertex types (1/15)(46)+(6/15)(42,52)+(2/15)(53)+(6/15)(52,4). The critical point is where the longer bonds (on both the lattice and dual lattice) have occupation probability p = 2 sin (π/18) = 0.347296... which is the bond percolation threshold on a triangular lattice, and the shorter bonds have occupation probability 1 - 2 sin(π/18) = 0.652703..., which is the bond percolation on a hexagonal lattice. These results follow from the isoradial condition [35] but also follow from applying the star-triangle transformation to certain stars on the honeycomb lattice. Finally, it can be generalized to having three different probabilities in the three different directions, p1, p2 and p3 for the long bonds, and 1 - p1, 1 - p2, and 1 - p3 for the short bonds, where p1, p2 and p3 satisfy the critical surface for the inhomogenous triangular lattice.


    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/medial lattice, same as the 2x2, 1x1 subnet for kagome-type lattices (removed).


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

    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[36] 0.707107... = 1/√2 [37]
    bow-tie (c) 3,4 3 1/7 0.672929..., 1-2p3-2p4-2p5-7p6+18p7+11p8-35p9+21p10-4p11=0 [38]
    bow-tie (d) 3,4 3⅓ 0.625457..., 1-2p2-3p3+4p4-p5=0 [38]
    martini-A (2/3)(3,72)+(1/3)(3,73) 3,4 3⅓ 1/√2[38] 0.625457..., 1-2p2-3p3+4p4-p5=0 [38]
    bow-tie dual lattice (e) 3,4 3⅔ 0.595482..., 1-pcbond (bow-tie (a)) [38]
    bow-tie (b) 3,4,6 3⅔ 0.533213..., 1-p- 2p3 -4p4-4p5+156+ 13p7-36p8+19p9+ p10 + p11=0 [38]
    martini covering/medial (1/2)(33,9)+(1/2)(3,9,3,9) 4 4 0.707107... = 1/√2 [37] 0.57086651(33) [39]
    martini-B (1/2)(3,5,3,52)+(1/2)(3,52) 3, 5 4 0.618034... = 2/(1 +√5)..., 1- p2-p=0[36][38] 1/2 [37][38]
    bow-tie dual lattice (f) 3,4,8 4 2/5 0.466787..., 1-pcbond (bow-tie (b))[38]
    bow-tie (a) (1/2)(32,4,32,4)+(1/2)(3,4,3) 4,6 5 0.5472(2) [40] 0.404518..., 1 - p - 6p2 +6p3-p5=0 [38]
    bow-tie dual lattice (h) 3,6,8 5 0.374543..., 1-pcbond(bow-tie (d))[38]
    bow-tie dual lattice (g) 3,6,10 5½ 0.327071..., 1-pcbond(bow-tie (c))[38]

    Thresholds on other 2D lattices

    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    (4, 6, 12) covering/medial 4 4 pcbond(4, 6, 12) = 0.693731... 0.5593140(2),[5] 0.559315(1)[41]
    (4, 82) covering/medial, square kagome 4 4 pcbond(4,82) = 0.676803... 0.544798005(8),[5] 0.54479793(34)[41]
    (34, 6) medial 4 4 0.5247495(5)[5]
    (3,4,6,4) medial 4 4 0.51276 [5]
    (32, 4, 3, 4) medial 4 4 0.512682929(8)[5]
    (33, 42) medial 4 4 0.51252459859(2)[5]
    square covering (non-planar) 6 6 1/2 0.3371(1)[42]
    square matching lattice (non-planar) 8 8 1 - pcsite(square) = 0.407253... 0.25036834(6)[11]

    (4, 6, 12) covering/medial lattice

    (4, 82) covering/medial lattice

    (3,122) covering/medial lattice (in light grey), equivalent to the kagome (2 x 2) subnet, and in black, the dual of these lattices.

    Thresholds on subnet lattices

    The 2 × 2 subnet is known as the "triangular kagome" lattice [43]

    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    checkerboard – 2 × 2 subnet 4,3 0.596303(1) [44]
    checkerboard – 4 × 4 subnet 4,3 0.633685(9) [44]
    checkerboard – 8 × 8 subnet 4,3 0.642318(5) [44]
    checkerboard – 16 × 16 subnet 4,3 0.64237(1) [44]
    checkerboard- 32 × 32 subnet 4,3 0.64219(2) [44]
    checkerboard – \infty subnet 4,3 0.642216(10) [44]
    kagome – 2 × 2 subnet = (3, 122) covering/medial 4 pcbond (3, 122) = 0.74042077... 0.600861966953(1),[45] 0.6008624(10),[12] 0.60086193(3)[9]
    kagome – 3 × 3 subnet 4 0.6193296(10),[12] 0.61933176(5),[9] 0.61933044(32)[46]
    kagome – 4 × 4 subnet 4 0.625365(3),[12] 0.62536424(7)[9]
    kagome – \infty subnet 4 0.628961(2) [12]
    kagome – (1 × 1):(2 × 2) subnet = martini covering/medial 4 pcbond(martini) = 1/√2 = 0.707107... 0.57086648(36) [39]
    kagome – (1 × 1):(3 × 3) subnet 4,3 0.728355596425196...[9] 0.58609776(37) [46]
    kagome – (1 × 1):(4 × 4) subnet 0.738348473943256...[9]
    kagome – (1 × 1):(5 × 5) subnet 0.743548682503071...[9]
    kagome – (1 × 1):(6 × 6) subnet 0.746418147634282...[9]
    kagome – (2 × 2):(3 × 3) subnet 0.61091770(30) [46]
    triangular – 2 × 2 subnet 6,4 0.471628788 [44]
    triangular – 3 × 3 subnet 6,4 0.509077793 [44]
    triangular – 4 × 4 subnet 6,4 0.524364822 [44]
    triangular – 5 × 5 subnet 6,4 0.5315976(10) [44]
    triangular – \infty subnet 6,4 0.53993(1) [44]

    Thresholds of dimers a square lattice

    system z Site Threshold
    unoriented dimers 4 0.5617 [47]
    parallel dimers 4 0.5683[47]

    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. [48]

    l (polymer length) z Bond Percolation
    1 4 0.5(exact) [49]
    2 4 0.47697(4)[49]
    4 4 0.44892(6) [49]
    8 4 0.41880(4)[49]

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

    k z Site Thresholds Bond Thresholds
    1 4 0.593(2) [50] 0.5009(2) [50]
    2 4 0.564(2) [50] 0.4859(2) [50]
    3 4 0.552(2) [50] 0.4732(2) [50]
    4 4 0.542(2) [50] 0.4630(2) [50]
    5 4 0.531(2) [50] 0.4565(2) [50]
    6 4 0.522(2) [50] 0.4497(2) [50]
    7 4 0.511(2) [50] 0.4423(2) [50]
    8 4 0.502(2) [50] 0.4348(2) [50]
    9 4 0.493(2) [50] 0.4291(2) [50]
    10 4 0.488(2) [50] 0.4232(2) [50]
    11 4 0.482(2) [50] 0.4159(2) [50]
    12 4 0.476(2) [50] 0.4114(2) [50]
    13 4 0.471(2) [50] 0.4061(2) [50]
    14 4 0.467(2) [50] 0.4011(2) [50]
    15 4 0.4011(2) [50] 0.3979(2) [50]

    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),[51] (3-{\sqrt {5}})/2[25]

    Thresholds for 2D continuum models

    System Φc ηc nc
    Disks of radius r 0.67634831(2),[52] 0.6763475(6),[53] 0.676339(4) [54] 1.12808737(6),[52] 1.128085(2),[53] 1.128059(12) [54] 1.436322(2),[53] 1.436289(16) [54]
    Ellipses, aspect ratio ε = 2 0.63 [55] 0.76 1.94
    Ellipses, ε = 5 0.455 [56] 0.607 3.864
    Ellipses, ε = 10 0.301 [56] 0.358 4.56
    Ellipses, ε = 20 0.178 [56] 0.196 4.99
    Ellipses, ε = 50 0.081 [56] 0.084 5.38
    Ellipses, ε = 100 0.0417 [56] 0.0426 5.42
    Ellipses, ε = 1000 0.0043 [56] 0.00431 5.5
    Aligned squares of side \ell 0.66674349(3),[52] 0.66653(1),[57] 0.6666(4)[58] 1.09884280(9),[52] 1.0982(3),[59] 1.098(1)[58] 1.09884280(9),[52] 1.0982(3),[59] 1.098(1)[58]
    Randomly oriented squares 0.62554075(4),[52] 0.6254(2)[58] 0.9822723(1),[52] 0.9819(6)[58] 0.982278(14) [60] 0.9822723(1),[52] 0.9819(6)[58] 0.982278(14) [60]
    Rectangles, ε = 1.1 0.624870(7) 0.980484(19) 1.078532(21) [60]
    Rectangles, ε = 2 0.590635(5) 0.893147(13) 1.786294(26) [60]
    Rectangles, ε = 3 0.5405983(34) 0.777830(7) 2.333491(22) [60]
    Rectangles, ε = 4 0.4948145(38) 0.682830(8) 2.731318(30) [60]
    Rectangles, ε = 5 0.4551398(31) 0.607226(6) 3.036130(28) [60]
    Rectangles, ε = 10 0.3233507(25) 0.3906022(37) 3.906022(37) [60]
    Rectangles, ε = 20 0.2048518(22) 0.2292268(27) 4.584535(54) [60]
    Rectangles, ε = 50 0.09785513(36) 0.1029802(4) 5.149008(20) [60]
    Rectangles, ε = 100 0.0523676(6) 0.0537886(6) 5.378856(60) [60]
    Rectangles, ε = 200 0.02714526(34) 0.02752050(35) 5.504099(69) [60]
    Rectangles, ε = 1000 0.00559424(6) 0.00560995(6) 5.609947(60) [60]
    Sticks of length \ell 5.6372858(6),[52] 5.63726(2) [61]
    Power-law disks, x=2.05 0.993(1) [62] 4.90(1) 0.0380(6)
    Power-law disks, x=2.25 0.8591(5) [62] 1.959(5) 0.06930(12)
    Power-law disks, x=2.5 0.7836(4) [62] 1.5307(17) 0.09745(11)
    Power-law disks, x=4 0.69543(6) [62] 1.18853(19) 0.18916(3)
    Power-law disks, x=5 0.68643(13) [62] 1.1597(3) 0.22149(8)
    Power-law disks, x=6 0.68241(8) [62] 1.1470(1) 0.24340(5)
    Power-law disks, x=7 0.6803(8) [62] 1.140(6) 0.25933(16)
    Power-law disks, x=8 0.67917(9) [62] 1.1368(5) 0.27140(7)
    Power-law disks, x=9 0.67856(12) [62] 1.1349(4) 0.28098(9)
    Voids around disks of radius r 0.159(2) [63]

    \eta _{c}=\pi r^{2}N/L^{2} equals critical total area for disks, where N is the number of objects and L is the system size.

    \eta _{c}=\pi abN/L^{2} for ellipses of semi-major and semi-minor axes of a and b, respectively. Aspect ratio \epsilon =a/b with a>b.

    \eta _{c}=\ell mN/L^{2} for rectangles of dimensions \ell and m. Aspect ratio \epsilon =\ell /m with \ell >m.

    \eta _{c}=\pi xN/(4L^{2}(x-2)) for power-law distributed disks with {\hbox{Prob(radius}}\geq R)=R^{{-x}}, R\geq 1.

    \phi _{c}=1-e^{{-\eta _{c}}} equals critical area fraction.

    n_{c}=\ell ^{2}N/L^{2} equals number of objects of maximum length \ell =2a per unit area.

    For ellipses, n_{c}=(4\epsilon /\pi )\eta _{c}

    For void percolation, \phi _{c}=e^{{-\eta _{c}}} is the critical void fraction.

    For more ellipse values, see [55]

    For more rectangle values, see [60]

    Thresholds on 2D random and quasi-lattices

    Left to right: (a) Voronoi diagram (solid lines) and its dual, the 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), (d) the Relative Neighborhood Graph (black lines) [64] superimposed on the Delaunay triangulation (black plus grey lines) for the same set of 128 uniformly distributed random points.


    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    Relative neighborhood graph 2.5576 0.796(2) [64] 0.771(2) [64]
    Voronoi tessellation 3 0.71410(2),[65] 0.7151* [33] 0.68,[66] 0.666931(5),[65] 0.6670(1) [67]
    Voronoi covering/medial 4 0.666931(2)[65][67] 0.53618(2) [65]
    Penrose rhomb dual 4 0.6381(3)[29] 0.5233(2) [29]
    Penrose rhomb 4 0.5837(3),[29] 0.58391(1)[68] 0.4770(2) [29]
    Delaunay triangulation 6 1/2 [69] 0.333069(2) [65][67]

    *Theoretical estimate

    Thresholds on slabs

    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    h= 2, SC, open b.c. 0.47424 [70]
    h = 3, BCC, periodic b.c. 0.21113018(38) [71]
    h = 4, BCC, periodic b.c. 0.20235168(59) [71]
    h= 4, SC, open b.c. 0.3997 [70]
    h = 5, SC, periodic b.c. 0.278102(5) [71]
    h = 6, SC, periodic b.c. 0.272380(2) [71]
    h = 7, SC, periodic b.c. 5,6 5,6 0.3459514(12) [71] 0.268459(1) [71]
    h= 8, SC, open b.c. 0.3557 [70]
    h = 8, SC, periodic b.c. 0.265615(5) [71]

    More for SC open b.c. in Ref.[70]

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

    Thresholds on 3D lattices

    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold Dimer Percolation Threshold
    (8,3)-a[72] 3 3 0.577962(33)[72] 0.555700(22)[72]
    (10,3)-a[72] 3 3 0.571404(40)[72] 0.551060(37)[72]
    (10,3)-b[72] 3 3 0.565442(40)[72] 0.546694(33)[72]
    ice 4 4 0.433(11)[73] 0.388(10)[74]
    diamond 4 4 0.4299870(4),[75] 0.426(+0.08,-0.02),[76] 0.4301(4)[77] 0.3895892(5),[75] 0.390(11),[74] 0.3893(2)[77]
    simple cubic 6 6 0.3116077(2),[78] 0.311604(6),[79] 0.311605(5),[80] 0.311600(5),[81] 0.3116077(4),[82] 0.3116081(13),[83] 0.3116080(4),[84] 0.3116004(35),[85] 0.31160768(15)[75] 0.24881182(10),[78] 0.2488125(25),[86]

    0.2488126(5) [87]

    		 0.2555(1)[88]
    Icosahedral Penrose 6 0.285[89] 0.225 [89]
    Penrose w/2 diagonals 6.764 0.271[89] 0.207 [89]
    Stacked triangular / simple hexagonal 8 8 0.26240(5),[90] 0.2625(2),[91] 0.2623(2)[40] 0.18602(2),[90] 0.1859(2) [40]
    bcc 8 8 0.2459615(10),[84] 0.2460(3),[92] 0.2464(7) [93] 0.1802875(10)[87]
    simple cubic with 3NN 8 8 0.2455(1) [94]
    fcc 12 12 0.1992365(10),[84] 0.19923517(20)[75] 0.1201635(10)[87]
    hcp 12 12 0.1992555(10)[95] 0.1201640(10)[95]
    La2-x Srx Cu O4 12 12 0.19927(2) [96]
    simple cubic with 2NN 12 12 0.1991(1) [94]
    Penrose w/8 diagonals 12.764 0.188[89] 0.111 [89]
    simple cubic with NN+3NN 14 14 0.1420(1) [94]
    simple cubic with NN+2NN 18 18 0.1372(1),[94] 0.13735(5) [97]
    simple cubic with short-length correlation 6+ 6+ 0.126(1)[98]
    simple cubic with 2NN+3NN 20 20 0.1036(1) [94]
    simple cubic with NN+2NN+3NN 26 26 0.0976(1),[94] 0.0976445(10) [97]

    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?

    Thresholds for 3D continuum models

    All overlapping except for jammed spheres.

    System Φc ηc
    Spheres of radius r 0.289573(2) [99] 0.341889(3) [99]
    Aligned cylinders 0.2819(2)[100] 0.3312(1)[100]
    Aligned cubes of side \ell =2a 0.2773(2) [58] 0.3247(3),[59] 0.3248(3)[58]
    Randomly oriented icosahedra 0.3030(5) [57]
    Randomly oriented dodecahedra 0.2949(5) [57]
    Randomly oriented octahedra 0.2514(6) [57]
    Randomly oriented cubes of side \ell =2a 0.2168(2) [58] 0.2444(3),[58] 0.2443(5)[57]
    Randomly oriented tetrahedra 0.1701(7) [57]
    Randomly oriented disks of radius r (in 3D) 0.9614(5)[101]
    Randomly oriented square plates of side {\sqrt {\pi }}r 0.8647(6)[101]
    Randomly oriented triangular plates of side {\sqrt {2\pi }}/3^{{1/4}}r 0.7295(6)[101]
    Voids around disks of radius r 22.86(2)[102]
    Voids around oblate ellipsoids of major radius r and aspect ratio 10 15.42(1)[102]
    Voids around oblate ellipsoids of major radius r and aspect ratio 2 6.478(8)[102]
    Voids around spheres of radius r 0.030(2),[63] 0.0301(3),[103] 0.0294,[104] 0.0300(3) [105] 3.506(8),[105] 3.515(6) [102]
    Jammed spheres (average z = 6) 0.183(3)[106]

    \eta _{c}=(4/3)\pi r^{3}N/L^{3} 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.

    For more results on void percolation around ellipsoids and elliptical plates, see.[102]

    Thresholds on hypercubic lattices

    d z Site Thresholds Bond Thresholds
    4 8 0.1968861(14),[107] 0.196889(3),[108] 0.196901(5) [109] 0.1601314(13),[107] 0.160130(3),[108] 0.1601310(10) [86]
    5 10 0.1407966(15) [107] 0.118172(1),[107] 0.1181718(3) [86]
    6 12 0.109017(2) [107] 0.0942019(6) [107]
    7 14 0.0889511(9),[107] 0.088939(20) [110] 0.0786752(3) [107]
    8 16 0.0752101(5) [107] 0.06770839(7) [107]
    9 18 0.0652095(3) [107] 0.05949601(5) [107]
    10 20 0.0575930(1) [107] 0.05309258(4) [107]
    11 22 0.05158971(8) [107] 0.04794969(1) [107]
    12 24 0.04673099(6) [107] 0.04372386(1) [107]
    13 26 0.04271508(8) [107] 0.04018762(1) [107]
    d z Site Thresholds Bond Thresholds τ
    4 8 0.196889(3) [108] 0.160130(3) [108] 2.313(3) [108]
    5 10 0.14081(1) [108] 0.118174(4) [108] 2.412(4) [108]

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

    d z Site Thresholds Bond Thresholds zspread dmin
    4 8 0.196889 [108] 0.160130 [108] 0.622(2) [108] 1.607(5) [108]
    5 10 0.14081 [108] 0.118174 [108] 0.552(2) [108] 1.812(6) [108]

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

    Thresholds on kagome lattices in higher dimensions

    d z Site Thresholds Bond Thresholds rw
    3 6 0.3895(2) [111] 0.417(1) [111]
    4 8 0.2715(3) [111] 0.274(1) [111]
    5 10 0.2084(4) [111] 0.208(1) [111]
    6 12 0.1677(7) [111] 0.170(1) [111]

    Thresholds on hyperbolic, hierarchical, and tree lattices

    Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk [112]


    Depiction of the non-planar Hanoi network HN-NP [113]


    Lattice z \overline z Site Percolation Threshold Bond Percolation Threshold
    Lower Upper
    {4,5} hyperbolic 5 5 0.27[114] 0.52[114]
    {7,3} hyperbolic 3 3 0.72[114] 0.53[114]
    {3,7} hyperbolic 7 7 0.20[114] 0.37[114]
    {∞,3} Cayley tree 3 3 1/2 1/2[114] 1[114]
    Enhanced binary tree (EBT) 0.304(1)[114] 0.48,[114] 0.564(1)[115]
    Enhanced binary tree dual 0.436(1)[115] 0.696(1)[115]
    Non-Planar Hanoi Network (HN-NP) 0.319445[113] 0.381996[113]
    Cayley tree with grandparents 8 0.158656326[116]

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

    Cayley tree (Bethe latttice) with coordination number z: pc= 1 / (z - 1)

    Cayley tree with a distribution of z with mean \overline z, mean-square \overline {z^{2}}: pc= \overline z/(\overline {z^{2}}-\overline z) [117] (site or bond threshold)

    Thresholds for directed percolation

    Lattice z Site Percolation Threshold Bond Percolation Threshold
    (1+1)-d honeycomb 1.5 0.8399316(2),[118] 0.839933(5),[119] 0.8228569(2),[118] 0.82285680(6)[118]
    (1+1)-d kagome 2 0.7369317(2),[118] 0.73693182(4)[120] 0.6589689(2),[118] 0.65896910(8)[118]
    (1+1)-d square, diagonal direction 2 0.705489(4),[121] 0.70548522(4),[122] 0.70548515(20),[120]

    0.7054852(3),[118]

    		 0.644701(2),[123] 0.644701(1),[124] 0.64470015(5),[125] 0.644700185(5),[122] 0.6447001(2),[118]
    (1+1)-d triangular 3 0.5956468(5),[125] 0.5956470(3) [118] 0.478025(1),[125] 0.4780250(4) [118]
    (2+1)-d simple cubic, diagonal planes 3 0.43531(1) [126] 0.382223(7) [126]
    (2+1)-d square nn (= bcc) 4 0.3445736(3),[127] 0.344575(15) [128] 0.2873383(1),[129] 0.287338(3)[126]
    (3+1)-d hypercubic, diagonal planes 4 0.3025(10) [130]
    (3+1)-d cubic, nn 6 0.2081040(4) [127] 0.1774970(5) [86]
    (3+1)-d body-centered hypercubic 8 0.160950(30) [128]
    (4+1)-d hypercubic, nn 8 0.1461593(2),[127] 0.1461582(3) [131] 0.1288557(5) [86]
    (4+1)-d body-centered hypercubic 16 0.075582(17) [128]

    0.0755850(3) [131]
    (5+1)-d hypercubic, nn 10 0.1123373(2) [127] 0.1016796(5) [86]
    (5+1)-d body-centered hypercubic 32 0.035967(23) [128]
    (6+1)-d hypercubic, nn 12 0.0913087(2) [127] 0.0841997(14) [86]
    (7+1)-d hypercubic,nn 14 0.07699336(7) [127] 0.07195(5) [86]

    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.

    General formulas for exact results

    Inhomogeneous triangular lattice bond percolation[13]

    1-p_{1}-p_{2}-p_{3}+p_{1}p_{2}p_{3}=0

    Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation[13]

    1-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}+p_{1}p_{2}p_{3}=0

    Inhomogeneous (3,12^2) lattice, site percolation[4] [132]

    1-3(s_{1}s_{2})^{2}+(s_{1}s_{2})^{3}=0, or s_{1}s_{2}=1-2\sin(\pi /18)

    Inhomogeneous martini lattice, bond percolation [38] 1-(p_{1}p_{2}r_{3}+p_{2}p_{3}r_{1}+p_{1}p_{3}r_{2})-(p_{1}p_{2}r_{1}r_{2}+p_{1}p_{3}r_{1}r_{3}+p_{2}p_{3}r_{2}r_{3})+p_{1}p_{2}p_{3}(r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3})+ r_{1}r_{2}r_{3}(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})-2p_{1}p_{2}p_{3}r_{1}r_{2}r_{3}=0

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

    1-r(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3}-p_{1}p_{2}p_{3})=0

    Inhomogeneous martini-A (3–7) lattice, bond percolation. Left side (top of "A" to bottom): r_{2},\ p_{1}. Right side: r_{1},\ p_{2}. Cross bond: \ r_{3}.

    1-p_{1}r_{2}-p_{2}r_{1}-p_{1}p_{2}r_{3}-p_{1}r_{1}r_{3}-p_{2}r_{2}r_{3}+p_{1}p_{2}r_{1}r_{3}+p_{1}p_{2}r_{2}r_{3}+p_{1}r_{1}r_{2}r_{3}+p_{2}r_{1}r_{2}r_{3}-p_{1}p_{2}r_{1}r_{2}r_{3}=0

    Inhomogeneous martini-B (3–5) lattice, bond percolation

    Inhomogeneous checkerboard lattice, bond percolation [26][51]

    1-(p_{1}p_{2}+p_{1}p_{3}+p_{1}p_{4}+p_{2}p_{3}+p_{2}p_{4}+p_{3}p_{4})+p_{1}p_{2}p_{3}+p_{1}p_{2}p_{4}+p_{1}p_{3}p_{4}+p_{2}p_{3}p_{4}=0

    Inhomogeneous bow-tie lattice, bond percolation [25][51]

    1-(p_{1}p_{2}+p_{1}p_{3}+p_{1}p_{4}+p_{2}p_{3}+p_{2}p_{4}+p_{3}p_{4})+p_{1}p_{2}p_{3}+p_{1}p_{2}p_{4}+p_{1}p_{3}p_{4}+p_{2}p_{3}p_{4}+ u(1-p_{1}p_{2}-p_{3}p_{4}+p_{1}p_{2}p_{3}p_{4})=0

    where p_{1},p_{2},p_{3},p_{4} are the four bonds around the square and u is the diagonal bond connecting the vertex between bonds p_{4},p_{1} and p_{2},p_{3}.

    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 Erdős–Rényi (ER) graphs with Poissonian degree distribution, pc=1/<k>.[133] The critical threshold was calculated exactly also for interdependent ER networks.[134]

    See also

    References

    1. ↑ Kasteleyn, P. W.; C. M. Fortuin (1969). "Phase transitions in lattice systems with random local properties". Journal of the Physical Society of Japan (Supplements) 26: 11–14. 
    2. ↑ 2.0 2.1 2.2 2.3 2.4 Grünbaum, Branko; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. 
    3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 Parviainen, Robert (2005). Connectivity Properties of Archimedean and Laves Lattices 34. Uppsala Dissertations in Mathematics. p. 37. ISBN 91-506-1751-6. 
    4. ↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Suding, P. N.; R. M. Ziff (1999). "Site percolation thresholds for Archimedean lattices". Physical Review E 60 (1): 275–283. Bibcode:1999PhRvE..60..275S. doi:10.1103/PhysRevE.60.275. 
    5. ↑ 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 Jacobsen, J. L. (2014). High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials. arXiv:1401.7847. 
    6. ↑ 6.0 6.1 Jacobsen, Jesper L.; Christian R. Scullard (2013). "Critical manifolds, graph polynomials, and exact solvability". StatPhys 25, Seoul, Korea July 21–26 http://www.statphys25.org/data/Statphys25%20Abstract%20Book.pdf. 
    7. ↑ 7.0 7.1 Scullard, C. R.; J. L. Jacobsen (2012). Transfer matrix computation of generalised critical polynomials in percolation. arXiv:1209.1451. Bibcode:2012arXiv1209.1451S. 
    8. ↑ 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 Parviainen, Robert (2007). "Estimation of bond percolation thresholds on the Archimedean lattices". J. Phys. A 40 (31): 9253–9258. arXiv:0704.2098. Bibcode:2007JPhA...40.9253P. doi:10.1088/1751-8113/40/31/005. 
    9. ↑ 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 Ding, Chengxiang; Zhe Fu. Wenan Guo, F. Y. Wu (2010). "Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis". Physical Review E 81 (6): 061111. arXiv:1001.1488. Bibcode:2010PhRvE..81f1111D. doi:10.1103/PhysRevE.81.061111. 
    10. ↑ Djordjevic, Z. V.; H. E. Stanley, Alla Margolina (1982). "Site percolation threshold for honeycomb and square lattices". J. Phys. A: Math. Gen. 15: L405–L412. doi:10.1088/0305-4470/15/8/006. 
    11. ↑ 11.0 11.1 11.2 11.3 11.4 Feng, Xiaomei; Youjin Deng; Henk W. J. Blote (2008). "Percolation transitions in two dimensions". Physical Review E 78 (3): 031136. Bibcode:2008PhRvE..78c1136F. doi:10.1103/PhysRevE.78.031136. 
    12. ↑ 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 Ziff, R. M.; Hang Gu (2008). Universal relation for critical percolation thresholds of kagome-class lattices. 
    13. ↑ 13.0 13.1 13.2 13.3 13.4 Sykes, M. F.; J. W. Essam (1964). "Exact critical percolation probabilities for site and bond problems in two dimensions". Journal of Mathematical Physics 5 (8): 1117–1127. Bibcode:1964JMP.....5.1117S. doi:10.1063/1.1704215. 
    14. ↑ Scullard, C. R. (2012). "Percolation critical polynomial as a graph invariant". Physical Review E 86 (4). arXiv:1111.1061. Bibcode:2012PhRvE..86d1131S. doi:10.1103/PhysRevE.86.041131. 
    15. ↑ 15.0 15.1 Ziff, R. M.; P. W. Suding (1997). "Determination of the bond percolation threshold for the kagome lattice". Journal of Physics A 30 (15): 5351–5359. arXiv:cond-mat/9707110. Bibcode:1997JPhA...30.5351Z. doi:10.1088/0305-4470/30/15/021. 
    16. ↑ Lin, Keh Ying; Wen Jong Ma (1983). "Two-dimensional Ising model on a ruby lattice". Journal of Physics A 16 (16): 3895–3898. Bibcode:1983JPhA...16.3895L. doi:10.1088/0305-4470/16/16/027. 
    17. ↑ Newman, M. E. J.; R. M. Ziff (2000). "Efficient Monte-Carlo algorithm and high-precision results for percolation". Physical Review Letters 85 (19): 4104–7. arXiv:cond-mat/0005264. Bibcode:2000PhRvL..85.4104N. doi:10.1103/PhysRevLett.85.4104. PMID 11056635. 
    18. ↑ de Oliveira, P.M.C.; R. A. Nobrega, D. Stauffer. (2003). "Corrections to finite size scaling in percolation". Brazilian Journal of Physics 33 (3): 616–618. doi:10.1590/S0103-97332003000300025. 
    19. ↑ Lee, M. J. (2007). "Complementary algorithms for graphs and percolation". Physical Review E 76 (2): 027702. arXiv:0708.0600. Bibcode:2007PhRvE..76b7702L. doi:10.1103/PhysRevE.76.027702. 
    20. ↑ Lee, M. J. (2008). "Pseudo-random-number generators and the square site percolation threshold". Physical Review E 78 (3): 031131. arXiv:0807.1576. Bibcode:2008PhRvE..78c1131L. doi:10.1103/PhysRevE.78.031131. 
    21. ↑ Betts, D. D. (1995). "A new two-dimensional lattice of coordination number five". Proc. Nova Scotian. Inst. Sci. 40: 95–100. 
    22. ↑ 22.0 22.1 22.2 22.3 Malarz, K.; S. Galam (2005). "Square-lattice site percolation at increasing ranges of neighbor bonds". Physical Review E 71 (1): 016125. arXiv:cond-mat/0408338. Bibcode:2005PhRvE..71a6125M. doi:10.1103/PhysRevE.71.016125. 
    23. ↑ 23.0 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 23.10 23.11 23.12 23.13 23.14 23.15 23.16 23.17 23.18 23.19 23.20 23.21 23.22 23.23 23.24 23.25 Majewski, M.; K. Malarz (2007). "Square lattice site percolation thresholds for complex neighbourhoods". Acta Phys. Pol. B 38 (38): 2191. arXiv:cond-mat/0609635. Bibcode:2007AcPPB..38.2191M. 
    24. ↑ Collier, Andrew. "Percolation Threshold: Including Next-Nearest Neighbours". 
    25. ↑ 25.0 25.1 25.2 Scullard, C. R.; R. M. Ziff (2010). "Critical surfaces for general inhomogeneous bond percolation problems". J. Stat. Mech: Th. Exp 2010 (3): P03021. arXiv:0911.2686. Bibcode:2010JSMTE..03..021S. doi:10.1088/1742-5468/2010/03/P03021. 
    26. ↑ 26.0 26.1 Wu, F. Y. (1979). "Critical point of planar Potts models". Journal of Physics C 12 (17): L645–L650. Bibcode:1979JPhC...12L.645W. doi:10.1088/0022-3719/12/17/002. 
    27. ↑ Scullard, C. R. (2010). To be published. 
    28. ↑
    29. ↑ 29.0 29.1 29.2 29.3 29.4 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. Bibcode:1989JPhA...22L.699S. doi:10.1088/0305-4470/22/14/009. 
    30. ↑ Deng, Y.; Y. Huang, J. L. Jacobsen, J. Salas, and A. D. Sokal, (2006). "Finite-temperature phase transition in a class of four-state Potts antiferromagnets". Physical Review Letters 107: 150601. arXiv:1108.1743. Bibcode:2011PhRvL.107o0601D. doi:10.1103/PhysRevLett.107.150601. 
    31. ↑ Syozi, I (1972). Domb, C; Green, M. S., eds. Phase Transitions in Critical Phenomena. Academic Press, London. pp. 270–329  Missing or empty |title= (help)
    32. ↑ 32.0 32.1 32.2 32.3 32.4 32.5 32.6 Hovi, J.-P.; A. Aharony (1996). "Scaling and universality in the spanning probability for percolation". Physical Review E 53 (1): 235–253. Bibcode:1996PhRvE..53..235H. doi:10.1103/PhysRevE.53.235. 
    33. ↑ 33.0 33.1 33.2 33.3 33.4 33.5 33.6 33.7 33.8 33.9 33.10 33.11 33.12 33.13 33.14 33.15 33.16 33.17 33.18 33.19 33.20 Neher, Richard; Mecke, Klaus and Wagner, Herbert (2008). "Topological estimation of percolation thresholds". Journal of Statistical Mechanics: Theory and Experiment 2008: P01011. arXiv:0708.3250. Bibcode:2008JSMTE..01..011N. doi:10.1088/1742-5468/2008/01/P01011. 
    34. ↑ 34.0 34.1 34.2 34.3 34.4 34.5 34.6 34.7 34.8 34.9 34.10 34.11 34.12 34.13 34.14 34.15 34.16 34.17 34.18 34.19 Gu, Hang; R. M. Ziff (2007). "Percolation thresholds of 2-uniform lattice". To be published. 
    35. ↑ Grimmett, G.; Manolescu, I. Bond percolation on isoradial graphs. arXiv:1204.0505. Bibcode:2012arXiv1204.0505G. 
    36. ↑ 36.0 36.1 Scullard, C. R. (2006). "Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation". Physical Review E 73: 016107. arXiv:cond-mat/0507392. Bibcode:2006PhRvE..73a6107S. doi:10.1103/PhysRevE.73.016107. 
    37. ↑ 37.0 37.1 37.2 Ziff, R. M. (2006). "Generalized cell–dual-cell transformation and exact thresholds for percolation". Physical Review E 73: 016134. Bibcode:2006PhRvE..73a6134Z. doi:10.1103/PhysRevE.73.016134. 
    38. ↑ 38.0 38.1 38.2 38.3 38.4 38.5 38.6 38.7 38.8 38.9 38.10 38.11 38.12 Scullard, C. R.; Robert M Ziff (2006). "Exact bond percolation thresholds in two dimensions". Journal of Physics A 39 (49): 15089. arXiv:cond-mat/0610813. Bibcode:2006JPhA...3915083Z. doi:10.1088/0305-4470/39/49/003. 
    39. ↑ 39.0 39.1 Gu, Hang; A. Becker, R. M. Ziff (2009). "Percolation on the Voronoi covering/medial diagram and several other four-coordinated lattices". To be published. 
    40. ↑ 40.0 40.1 40.2 van der Marck, S. C. (1997). "Percolation thresholds and universal formulas". Physical Review E 55 (2): 1514–1517. Bibcode:1997PhRvE..55.1514V. doi:10.1103/PhysRevE.55.1514. 
    41. ↑ 41.0 41.1 Ziff, R. M. (2013). To be published. 
    42. ↑ Ziff, R. M.; Scullard, C. R. (2010). "Critical surfaces for general inhomogeneous bond percolation problems". J. Stat. Mech 2010 (3): P03021. arXiv:0911.2686. Bibcode:2010JSMTE..03..021S. doi:10.1088/1742-5468/2010/03/P03021. 
    43. ↑ Okubo, S.; M. Hayashi, S. Kimura, H. Ohta, M. Motokawa, H. Kikuchi and H. Nagasawa (1998). "Submillimeter wave ESR of triangular-kagome antiferromagnet Cu9X2(cpa)6 (X=Cl, Br)". Physica B. 246--247 (2): 553–556. Bibcode:1998PhyB..246..553O. doi:10.1016/S0921-4526(97)00985-X. 
    44. ↑ 44.0 44.1 44.2 44.3 44.4 44.5 44.6 44.7 44.8 44.9 44.10 Haji Akbari, Amir; R. M. Ziff (2009). "Percolation in networks with voids and bottlenecks". Physical Review E 79 (2): 021118. arXiv:0811.4575. Bibcode:2009PhRvE..79b1118H. doi:10.1103/PhysRevE.79.021118. 
    45. ↑
    46. ↑ 46.0 46.1 46.2 Gu, Hang; R. M. Ziff (2010). To be published. 
    47. ↑ 47.0 47.1 Cherkasova, V. A.; Yu. Yu. Tarasevich, N. I. Lebovka, and N.V. Vygornitskii (2010). "Percolation of the aligned dimers on a square lattice". Eur. Phys. J. B 74 (2): 205–209. arXiv:0912.0778. Bibcode:2010EPJB...74..205C. doi:10.1140/epjb/e2010-00089-2. 
    48. ↑ 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. 
    49. ↑ 49.0 49.1 49.2 49.3 Wu, Yong; B. Schmittmann, R. K. P. Zia (2008). "Two-dimensional polymer networks near percolation". Journal of Physics A 41 (2): 025008. Bibcode:2008JPhA...41b5004W. doi:10.1088/1751-8113/41/2/025004. 
    50. ↑ 50.0 50.1 50.2 50.3 50.4 50.5 50.6 50.7 50.8 50.9 50.10 50.11 50.12 50.13 50.14 50.15 50.16 50.17 50.18 50.19 50.20 50.21 50.22 50.23 50.24 50.25 50.26 50.27 50.28 50.29 Cornette, V.; A.J. Ramirez-Pastor, F. Nieto (2003). "Two-dimensional polymer networks near percolation". European Physical Journal B 36 (3): 397. Bibcode:2003EPJB...36..391C. doi:10.1140/epjb/e2003-00358-1. 
    51. ↑ 51.0 51.1 51.2 Ziff, R. M.; C. R. Scullard, J. C. Wierman, M. R. A. Sedlock (2012). "The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices". Journal of Physics A: Mathematical and Theoretical 45 (49): 494005. arXiv:1210.6609. Bibcode:2012JPhA...45W4005Z. doi:10.1088/1751-8113/45/49/494005. 
    52. ↑ 52.0 52.1 52.2 52.3 52.4 52.5 52.6 52.7 52.8 Mertens, Stephan; Cristopher Moore (2012). "Continuum percolation thresholds in two dimensions". Physical Review E 86: 061109. arXiv:1209.4936. Bibcode:2012PhRvE..86f1109M. doi:10.1103/PhysRevE.86.061109. 
    53. ↑ 53.0 53.1 53.2 Quintanilla, John A.; R. M. Ziff (2007). "Near symmetry of percolation thresholds of fully penetrable disks with two different radii". Physical Review E 76 (5): 051115 [6 pages]. Bibcode:2007PhRvE..76e1115Q. doi:10.1103/PhysRevE.76.051115. 
    54. ↑ 54.0 54.1 54.2 Quintanilla, J; S. Torquato; R. M. Ziff (2000). "Efficient measurement of the percolation threshold for fully penetrable discs". J. Phys. A: Math. Gen. 33 (42): L399–L407. Bibcode:2000JPhA...33L.399Q. doi:10.1088/0305-4470/33/42/104. 
    55. ↑ 55.0 55.1 Xia, W.; M. F. Thorpe (1988). "Percolation properties of random ellipses". Physical Review A 38 (5): 2650–2656. Bibcode:1988PhRvA..38.2650X. doi:10.1103/PhysRevA.38.2650. PMID 9900674. 
    56. ↑ 56.0 56.1 56.2 56.3 56.4 56.5 Yi, Y.-B.; A. M. Sastry (2007). "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution". Proceedings of the Royal Society A 460 (5): 2353–2380. Bibcode:2004RSPSA.460.2353Y. doi:10.1098/rspa.2004.1279. 
    57. ↑ 57.0 57.1 57.2 57.3 57.4 57.5 Torquato, S.; Y. Jiao (2012). Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles. arXiv:1210.0134. Bibcode:2013PhRvE..87b2111T. doi:10.1103/PhysRevE.87.022111. 
    58. ↑ 58.0 58.1 58.2 58.3 58.4 58.5 58.6 58.7 58.8 58.9 Baker, Don R.; Gerald Paul, Sameet Sreenivasan, H. Eugene Stanley (2002). "Continuum percolation threshold for interpenetrating squares and cubes". Physical Review E 66 (4): 046136 [5 pages]. arXiv:cond-mat/0203235. Bibcode:2002PhRvE..66d6136B. doi:10.1103/PhysRevE.66.046136. 
    59. ↑ 59.0 59.1 59.2 Torquato, S.; Y. Jiao (2012). "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses". J. Chem. Phys. 137 (7): 074106. arXiv:1208.3720. Bibcode:2012JChPh.137g4106T. doi:10.1063/1.4742750. 
    60. ↑ 60.0 60.1 60.2 60.3 60.4 60.5 60.6 60.7 60.8 60.9 60.10 60.11 60.12 60.13 Li, Jiantong; Mikael Östling (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles". Physical Review E 88 (1): 012101. Bibcode:2013PhRvE..88a2101L. doi:10.1103/PhysRevE.88.012101. 
    61. ↑ Li, Jiantong; Shi-Li Zhang (2009). "Finite-size scaling in stick percolation". Physical Review E 80 (4): 040104(R). Bibcode:2009PhRvE..80d0104L. doi:10.1103/PhysRevE.80.040104. 
    62. ↑ 62.0 62.1 62.2 62.3 62.4 62.5 62.6 62.7 62.8 Sasidevan, V. Continuum percolation of overlapping discs with a distribution of radii having a power-law tail. arXiv:1302.0085v2 cond-mat/ 1302.0085v2. 
    63. ↑ 63.0 63.1 van der Marck, S. C. (1996). "Network Approach to Void Percolation in a Pack of Unequal Spheres". Physical Review Letters 77 (9): 1785–1788. Bibcode:1996PhRvL..77.1785V. doi:10.1103/PhysRevLett.77.1785. PMID 10063171. 
    64. ↑ 64.0 64.1 64.2 Melchert, Oliver (2013). "Percolation thresholds on planar Euclidean relative-neighborhood graphs". Physical Review E 87 (4): 042106. arXiv:1301.6967. Bibcode:2013PhRvE..87d2106M. doi:10.1103/PhysRevE.87.042106. 
    65. ↑ 65.0 65.1 65.2 65.3 65.4 Becker, A.; R. M. Ziff (2009). "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations". Physical Review E 80 (4): 041101. arXiv:0906.4360. Bibcode:2009PhRvE..80d1101B. doi:10.1103/PhysRevE.80.041101. 
    66. ↑ Shante, K. S.; S. Kirkpatrick (1971). "An introduction to percolation theory". Advances in Physics 20 (85): 325–357. Bibcode:1971AdPhy..20..325S. doi:10.1080/00018737100101261. 
    67. ↑ 67.0 67.1 67.2 Hsu, H. P.; M. C. Huang (1999). "Percolation thresholds, critical exponents, and scaling functions on planar random lattices and their duals". Physical Review E 60 (1999): 6361–6370. Bibcode:1999PhRvE..60.6361H. doi:10.1103/PhysRevE.60.6361. 
    68. ↑ Ziff, R. M.; F. Babalievski (1999). "Site percolation on the Penrose rhomb lattice". Physica A 269 (2–4): 201–210. Bibcode:1999PhyA..269..201Z. doi:10.1016/S0378-4371(99)00166-1. 
    69. ↑ Bollobás, Béla; Oliver Riordan (2006). "The critical probability for random Voronoi percolation in the plane is 1/2". Probab. Theory Relat. Fields 136 (3): 417–468. doi:10.1007/s00440-005-0490-z. 
    70. ↑ 70.0 70.1 70.2 70.3 Sotta, P.; D. Long (2003). "The crossover from 2D to 3D percolation: Theory and numerical simulations". Eur. Phys. J. E 11 (4): 375–388. Bibcode:2003EPJE...11..375S. doi:10.1140/epje/i2002-10161-6. 
    71. ↑ 71.0 71.1 71.2 71.3 71.4 71.5 71.6 Gliozzi, F.; S. Lottini; M. Panero; A. Rago (2005). "Random percolation as a gauge theory". Nuclear Physics B 719 (3): 255–274. arXiv:cond-mat/0502339. Bibcode:2005NuPhB.719..255G. doi:10.1016/j.nuclphysb.2005.04.021. 
    72. ↑ 72.0 72.1 72.2 72.3 72.4 72.5 72.6 72.7 72.8 Tran, Jonathan; Ted Yoo, Shane Stahlheber, Alex Small (2012). Percolation thresholds on 3-dimensional lattices with 3 nearest neighbors. arXiv:1211.6531. Bibcode:2013JSMTE..05..014T. doi:10.1088/1742-5468/2013/05/P05014. 
    73. ↑ Vyssotsky, V. A.; H. L. Frisch; E. Sonnenblick; J. M. Hammersley (1961). "Critical Percolation Probabilities (Site Problem)". Physical Review 124 (4): 1021–1022. Bibcode:1961PhRv..124.1021F. doi:10.1103/PhysRev.124.1021. 
    74. ↑ 74.0 74.1 Vyssotsky, V. A.; S. B. Gordon; H. L. Frisch; J. M. Hammersley (1961). "Critical Percolation Probabilities (Bond Problem)". Physical Review 123 (5): 1566–1567. Bibcode:1961PhRv..123.1566V. doi:10.1103/PhysRev.123.1566. 
    75. ↑ 75.0 75.1 75.2 75.3 Xu, Xiao; Junfeng Wang, Jian-Ping Lv, Youjin Deng (2013). Simultaneous analysis of three-dimensional percolation models. arXiv:1310.5399v1. 
    76. ↑ Silverman, Amihal; J. Adler (1990). "Site-percolation threshold for a diamond lattice with diatomic substitution". Physical Review B 42 (2): 1369–1373. Bibcode:1990PhRvB..42.1369S. doi:10.1103/PhysRevB.42.1369. 
    77. ↑ 77.0 77.1 van der Marck, Steven C. (1998). "Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices". International Journal of Modern Physics C 9 (4): 529–540. arXiv:cond-mat/9802187. Bibcode:1998IJMPC...9..529V. doi:10.1142/S0129183198000431. 
    78. ↑ 78.0 78.1 Wang, J; Z. Zhou, W. Zhang, T. Garoni, Y. Deng (2013). Bond and site percolation in three dimensions. arXiv:1302.0421. Bibcode:2013PhRvE..87e2107W. doi:10.1103/PhysRevE.87.052107. 
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    80. ↑ Acharyya, M.; D. Stauffer (1998). "Effects of Boundary Conditions on the Critical Spanning Probability". Int. J. Mod. Phys. C 9 (4): 643–647. arXiv:cond-mat/9805355. Bibcode:1998IJMPC...9..643A. doi:10.1142/S0129183198000534. 
    81. ↑ Jan, N.; D. Stauffer (1998). "Random Site Percolation in Three Dimensions". Int. J. Mod. Phys. C 9 (4): 341–347. Bibcode:1998IJMPC...9..341J. doi:10.1142/S0129183198000261. 
    82. ↑ Deng, Youjin; H. W. J. Blöte (2005). "Monte Carlo study of the site-percolation model in two and three dimensions". Physical Review E 72 (1): 016126. Bibcode:2005PhRvE..72a6126D. doi:10.1103/PhysRevE.72.016126. 
    83. ↑ Ballesteros, P. N.; L. A. Fernández, V. Martín-Mayor, A. Muñoz, Sudepe, G. Parisi, and J. J. Ruiz-Lorenzo (1999). "Scaling corrections: site percolation and Ising model in three dimensions". Journal of Physics A 32: 1–13. arXiv:cond-mat/9805125. Bibcode:1999JPhA...32....1B. doi:10.1088/0305-4470/32/1/004. 
    84. ↑ 84.0 84.1 84.2 Lorenz, C. D.; R. M. Ziff (1998). "Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation". Journal of Physics A 31 (40): 8147–8157. arXiv:cond-mat/9806224. Bibcode:1998JPhA...31.8147L. doi:10.1088/0305-4470/31/40/009. 
    85. ↑ Škvor, Jiří; Ivo Nezbeda (2009). "Percolation threshold parameters of fluids". Physical Review E 79 (4): 041141. Bibcode:2009PhRvE..79d1141S. doi:10.1103/PhysRevE.79.041141. 
    86. ↑ 86.0 86.1 86.2 86.3 86.4 86.5 86.6 86.7 Dammer, Stephan M; Haye Hinrichsen (2004). "Spreading with immunization in high dimensions". J. Stat. Mech: Theory Exp. 2004 (7): P07011. arXiv:cond-mat/0405577. Bibcode:2004JSMTE..07..011D. doi:10.1088/1742-5468/2004/07/P07011. 
    87. ↑ 87.0 87.1 87.2 Lorenz, C. D.; R. M. Ziff (1998). "Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices". Physical Review E 57: 230–236. arXiv:cond-mat/9710044. Bibcode:1998PhRvE..57..230L. doi:10.1103/PhysRevE.57.230. 
    88. ↑ Tarasevich, Yu. Yu.; V. A. Cherkasova (2007). "Dimer percolation and jamming on simple cubic lattice". European Physical Journal B 60 (1): 97–100. arXiv:0709.3626. Bibcode:2007EPJB...60...97T. doi:10.1140/epjb/e2007-00321-2. 
    89. ↑ 89.0 89.1 89.2 89.3 89.4 89.5 Zakalyukin, R. M.; V. A. Chizhikov (2005). "Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method". Crystallography Reports 50 (6): 938–948. Bibcode:2005CryRp..50..938Z. doi:10.1134/1.2132400. 
    90. ↑ 90.0 90.1 Schrenk, K. J.; N. A. M. Araújo, H. J. Herrmann (2013). "Stacked triangular lattice: percolation properties". Physical Review E 87: 032123. arXiv:1302.0484. Bibcode:2013PhRvE..87c2123S. doi:10.1103/PhysRevE.87.032123. 
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    96. ↑ Tahir-Kheli, Jamil; W. A. Goddard III (2007). "Chiral plaquette polaron theory of cuprate superconductivity". Physical Review B 76: 014514. arXiv:0707.3535. Bibcode:2007PhRvB..76a4514T. doi:10.1103/PhysRevB.76.014514. 
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    98. ↑ Harter, T. (2005). "Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields". Physical Review E 72 (2): 026120. Bibcode:2005PhRvE..72b6120H. doi:10.1103/PhysRevE.72.026120. 
    99. ↑ 99.0 99.1 Lorenz, C. D.; R. M. Ziff (2000). "Precise determination of the critical percolation threshold for the three dimensional Swiss cheese model using a growth algorithm". J. Chem. Phys. 114 (8): 3659. Bibcode:2001JChPh.114.3659L. doi:10.1063/1.1338506. 
    100. ↑ 100.0 100.1 Hyytiä, E.; J. Virtamo, P. Lassila and J. Ott (2012). "Continuum Percolation Threshold for Permeable Aligned Cylinders and Opportunistic Networking". IEEE Communications Letters 16 (7): 1064–1067. doi:10.1109/LCOMM.2012.051512.120497. 
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    102. ↑ 102.0 102.1 102.2 102.3 102.4 Yi, Y. B.; K. Esmail (2012). "Computational measurement of void percolation thresholds of oblate particles and thin plate composites". J. Appl. Phys. 111: 124903. Bibcode:2012JAP...111l4903Y. doi:10.1063/1.4730333. 
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    104. ↑ Yi, Y. B. (2006). "Void percolation and conduction of overlapping ellipsoids". Physical Review E 74 (3): 031112. Bibcode:2006PhRvE..74c1112Y. doi:10.1103/PhysRevE.74.031112. 
    105. ↑ 105.0 105.1 Höfling, F.; T. Munk, E. Frey, and T. Franosch (2008). "Critical dynamics of ballistic and {B}rownian particles in a heterogeneous environment". J. Chem. Phys. 128 (16): 164517. arXiv:0712.2313. Bibcode:2008JChPh.128p4517H. doi:10.1063/1.2901170. 
    106. ↑ Powell, M. J. (1979). "Site percolation in randomly packed spheres". Physical Review B 20: 4194. Bibcode:1979PhRvB..20.4194P. doi:10.1103/PhysRevB.20.4194. 
    107. ↑ 107.0 107.1 107.2 107.3 107.4 107.5 107.6 107.7 107.8 107.9 107.10 107.11 107.12 107.13 107.14 107.15 107.16 107.17 107.18 107.19 Grassberger, Peter (2003). "Critical percolation in high dimensions". Physical Review E 67 (3): 4. arXiv:cond-mat/0202144. Bibcode:2003PhRvE..67c6101G. doi:10.1103/PhysRevE.67.036101. 
    108. ↑ 108.0 108.1 108.2 108.3 108.4 108.5 108.6 108.7 108.8 108.9 108.10 108.11 108.12 108.13 108.14 108.15 Paul, Gerald; Robert M. Ziff, H. Eugene Stanley (2001). "Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions". Physical Review E 64 (2): 8. arXiv:cond-mat/0101136. Bibcode:2001PhRvE..64b6115P. doi:10.1103/PhysRevE.64.026115. 
    109. ↑ Ballesteros, H. G.; L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo (1997). "Measures of critical exponents in the four dimensional site percolation". Phys. Lett. B 400 (3–4): 346–351. arXiv:hep-lat/9612024. Bibcode:1997PhLB..400..346B. doi:10.1016/S0370-2693(97)00337-7. 
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