Percolation threshold

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

Percolation models

The most common percolation model is to take a regular lattice, like a square lattice, and make it into a random network by randomly "occupying" sites or bonds with a statistically independent probability p. At a critical threshold p_c, large clusters and long-range connectivity first appear, and this is called the percolation threshold. Depending on the method for obtaining the random network, one distinguishes between the site percolation threshold and the bond percolation threshold. More general systems have several probabilities p₁, p₂, etc., and the transition is characterized by a critical surface or manifold. One can also consider continuum systems, such as overlapping disks and spheres placed randomly, or the negative space.
To understand the threshold, you can consider a quantity such as the probability that there is a continuous path from one boundary to another along occupied sites or bonds—that is, within a single cluster. For example, one can consider a square system, and ask for the probability P that there is a path from the top boundary to the bottom boundary. As a function of the occupation probability p, one finds a sigmoidal plot that goes from P=0 at p=0 to P=1 at p=1. The larger the square is compared to the lattice spacing, the sharper the transition will be. When the system size goes to infinity, P will be a step function at the threshold value p_c. For finite large systems, P is a constant whose value depends upon the shape of the system; for the square system discussed above, P= exactly for any lattice by a simple symmetry argument.
There are other signatures of the critical threshold. For example, the size distribution drops off as a power-law for large s at the threshold, n_s ~ s^−τ, where τ is a dimension-dependent percolation critical exponents. For an infinite system, the critical threshold corresponds to the first point where the size of the clusters become infinite.
In the systems described so far, it has been assumed that the occupation of a site or bond is completely random—this is the so-called Bernoulli percolation. For a continuum system, random occupancy corresponds to the points being placed by a Poisson process. Further variations involve correlated percolation, such as percolation clusters related to Ising and Potts models of ferromagnets, in which the bonds are put down by the Fortuin–Kasteleyn method. In bootstrap or k-sat percolation, sites and/or bonds are first occupied and then successively culled from a system if a site does not have at least k neighbors. Another important model of percolation, in a different universality class altogether, is directed percolation, where connectivity along a bond depends upon the direction of the flow. Another variation of recent interest is Explosive Percolation, whose thresholds are listed on that page.
Over the last several decades, a tremendous amount of work has gone into finding exact and approximate values of the percolation thresholds for a variety of these systems. Exact thresholds are only known for certain two-dimensional lattices that can be broken up into a self-dual array, such that under a triangle-triangle transformation, the system remains the same. Studies using numerical methods have led to numerous improvements in algorithms and several theoretical discoveries.
Simple duality in two dimensions implies that all fully triangulated lattices all have site thresholds of, and self-dual lattices have bond thresholds of.
The notation such as comes from Grünbaum and Shephard, and indicates that around a given vertex, going in the clockwise direction, one encounters first a square and then two octagons. Besides the eleven Archimedean lattices composed of regular polygons with every site equivalent, many other more complicated lattices with sites of different classes have been studied.
Error bars in the last digit or digits are shown by numbers in parentheses. Thus, 0.729724 signifies 0.729724 ± 0.000003, and 0.74042195 signifies 0.74042195 ± 0.00000080. The error bars variously represent one or two standard deviations in net error, or an empirical confidence interval, depending upon the source.

Percolation on networks

For a random tree-like network without degree-degree correlation, it can be shown that such network can have a giant component, and the percolation threshold is given by
Where is the generating function corresponding to the excess degree distribution, is the average degree of the network and is the second moment of the degree distribution. So, for example, for an ER network, since the degree distribution is a Poisson distribution, where the threshold is at.
In networks with low clustering,, the critical point gets scaled by such that:
This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold, mainly because for a fixed number of links, the clustering structure reinforces the core of the network with the price of diluting the global connections. For networks with high clustering, strong clustering could induce the core–periphery structure, in which the core and periphery might percolate at different critical points, and the above approximate treatment is not applicable.

Percolation in 2D

Thresholds on Archimedean lattices

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

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

2D lattices with extended and complex neighborhoods

In this section, sq-1,2,3 corresponds to square, etc. Equivalent to square-2N+3N+4N, sq. tri = triangular, hc = honeycomb.

Lattice	z	Site percolation threshold	Bond percolation threshold
sq-1, sq-2, sq-3, sq-5	4	0.5927...
sq-1,2, sq-2,3, sq-3,5... 3x3 square	8	0.407...	0.25036834, 0.2503685, 0.25036840
sq-1,3	8	0.337	0.2214995
sq-2,5: 2NN+5NN	8	0.337
hc-1,2,3: honeycomb-NN+2NN+3NN	12	0.300, 0.300, 0.302960... = 1-p_c
tri-1,2: triangular-NN+2NN	12	0.295, 0.289, 0.290258
tri-2,3: triangular-2NN+3NN	12	0.232020, 0.232020
sq-4: square-4NN	8	0.270...
sq-1,5: square-NN+5NN	8	0.277
sq-1,2,3: square-NN+2NN+3NN	12	0.292, 0.290 0.289, 0.288, 0.2891226	0.1522203
sq-2,3,5: square-2NN+3NN+5NN	12	0.288
sq-1,4: square-NN+4NN	12	0.236
sq-2,4: square-2NN+4NN	12	0.225
tri-4: triangular-4NN	12	0.192450, 0.1924428
hc-2,4: honeycomb-2NN+4NN	12	0.2374	-
tri-1,3: triangular-NN+3NN	12	0.264539
tri-1,2,3: triangular-NN+2NN+3NN	18	0.225, 0.215, 0.215459 0.2154657
sq-3,4: 3NN+4NN	12	0.221
sq-1,2,5: NN+2NN+5NN	12	0.240	0.13805374
sq-1,3,5: NN+3NN+5NN	12	0.233
sq-4,5: 4NN+5NN	12	0.199
sq-1,2,4: NN+2NN+4NN	16	0.219
sq-1,3,4: NN+3NN+4NN	16	0.208
sq-2,3,4: 2NN+3NN+4NN	16	0.202
sq-1,4,5: NN+4NN+5NN	16	0.187
sq-2,4,5: 2NN+4NN+5NN	16	0.182
sq-3,4,5: 3NN+4NN+5NN	16	0.179
sq-1,2,3,5 asterisk pattern	16	0.208	0.1032177
tri-4,5: 4NN+5NN	18	0.140250,
sq-1,2,3,4: NN+2NN+3NN+4NN	20	0.19671, 0.196, 0.196724, 0.1967293	0.0841509
sq-1,2,4,5: NN+2NN+4NN+5NN	20	0.177
sq-1,3,4,5: NN+3NN+4NN+5NN	20	0.172
sq-2,3,4,5: 2NN+3NN+4NN+5NN	20	0.167
sq-1,2,3,5,6 asterisk pattern	20		0.0783110
sq-1,2,3,4,5: NN+2NN+3NN+4NN+5NN	24	0.164, 0.164, 0.1647124
sq-1,2,3,4,6: NN+2NN+3NN+4NN+6NN	24	0.16134,
tri-1,4,5: NN+4NN+5NN	24	0.131660
sq-1,...,6: NN+...+6NN	28	0.142, 0.1432551	0.0558493
tri-2,3,4,5: 2NN+3NN+4NN+5NN	30	0.117460 0.135823
tri-1,2,3,4,5: NN+2NN+3NN+4NN+5NN	36	0.115, 0.115740, 0.1157399
sq-1,...,7: NN+...+7NN	36	0.113, 0.1153481	0.04169608
sq lat, diamond boundary: dist. ≤ 4	40	0.105
sq-1,...,8: NN+..+8NN	44	0.095, 0.095765, 0.09580, 0.0957661
sq-1,...,9: NN+..+9NN	48	0.086	0.02974268
sq-1,...,11: NN+...+11NN	60		0.02301190
sq-1,...,23	148		0.008342595
sq-1,...,32: NN+...+32NN	224		0.0053050415
sq-1,...,86: NN+...+86NN	708		0.001557644
sq-1,...,141: NN+...+141NN	1224		0.000880188
sq-1,...,185: NN+...+185NN	1652		0.000645458
sq-1,...,317: NN+...+317NN	3000		0.000349601
sq-1,...,413: NN+...+413NN	4016		0.0002594722
sq lat, diamond boundary: dist. ≤ 6	84	0.049
sq lat, diamond boundary: dist. ≤ 8	144	0.028
sq lat, diamond boundary: dist. ≤ 10	220	0.019
2x2 touching lattice squares*	20	φ_c = 0.58365, p_c = 0.196724, 0.19671,	-
3x3 touching lattice squares* )	44	φ_c = 0.59586, p_c = 0.095765, 0.09580
4x4 touching lattice squares*	76	φ_c = 0.60648, p_c = 0.0566227, 0.05665,
5x5 touching lattice squares*	116	φ_c = 0.61467, p_c = 0.037428, 0.03745,
6x6 touching lattice squares*	220	p_c = 0.02663,
10x10 touching lattice squares*	436	φ_c = 0.63609, p_c = 0.0100576
within 11 x 11 square	120		0.01048079
within 15 x 15 square	224		0.005287692
20x20 touching lattice squares*	1676	φ_c = 0.65006, p_c = 0.0026215
within 31 x 31 square	960		0.001131082
100x100 touching lattice squares*	40396	φ_c = 0.66318, p_c = 0.000108815
1000x1000 touching lattice squares*	4003996	φ_c = 0.66639, p_c = 1.09778E-06

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

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