Percolation threshold

Percolation threshold is a mathematical term related to percolation theory, which is the formation of long-range connectivity in random systems. In engineering and coffee making, percolation is the slow flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems, and the nature of the connectivity on them. The percolation threshold is the critical value of the occupation probability"p", or more generally a critical surface for a group of parameters "p"₁, "p"₂, ...,such that infinite connectivity ("percolation") first occurs.

Percolation models

The most common percolation model is to take a regular lattice, like a square lattice, andmake it into a random network by randomly "occupying" sites (vertices) or bonds (edges) with astatistically independent probability "p". At a critical threshold "p_c", long-rangeconnectivity first appears, and this is called the percolation threshold. More general systems have several probabilities "p"₁, "p"₂, 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 negativespace ("Swiss-cheese" models).

In the systems described so far, it has been assumed that the occupationof 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 whichthe bonds are put down by the Fortuin-Kasteleyn method.cite journal
last = Kasteleyn
first = P. W.
authorlink =
coauthors = C. M. Fortuin
title = Phase transitions in lattice systems with random local properties
journal = Journal of the Physical Society of Japan (Supplements)
volume = 26
issue =
year = 1969
pages = 11–14] In "bootstrap" or"k-sat" percolation, sites and/or bonds are first occupied and thensuccessively 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 intofinding exact and approximate values of the percolation thresholds for a varietyof these systems. Exact thresholds are only known for certain two-dimensionallattices that can be broken up into a self-dual array, such that under atriangle-triangle transformation, the system remains the same. Studiesusing numerical methods have led to numerous improvements in algorithms andseveral theoretical discoveries.

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

The notation such as (4,8²) comes from Grünbaum and Shepard,cite book
author=Grünbaum, Branko; and Shephard, G. C.
title=Tilings and Patterns
location=New York
publisher=W. H. Freeman
year=1987
id=ISBN 0-716-71193-1] and indicates that around a given vertex, going in the clockwise direction, one encountersfirst a square and then two octagons. Besides the eleven Archimedean latticescomposed of regular polygons with every site equivalent, many other more complicated lattices with sites of different classes have been studied.

Thresholds on 2d regular and Archimedean lattices

Thresholds on other 2d lattices

Thresholds on 2d random and quasi-lattices

General formulas for exact results

Inhomogeneous triangular lattice

Inhomogeneous honeycomb lattice

Inhomogeneous martini latticecite journal
last = Scullard
first = Christian R.
authorlink =
coauthors = R. M. Ziff
title = Predictions of bond percolation thresholds for the kagomé and Archimedean (3,12²) lattices
journal = Physical Review E
volume = 73
issue =
year = 2006
pages = 045102R
doi = 10.1103/PhysRevE.73.045102]

$1\; -\; (p\_1\; p\_2\; r\_3\; +\; p\_2\; p\_3\; r\_1\; +\; p\_1\; p\_3\; r\_2)\; -\; (p\_1\; p\_2\; r\_1\; r\_2\; +\; p\_1\; p\_3\; r\_1\; r\_3\; +\; p\_2\; p\_3\; r\_2\; r\_3)\; +\; p\_1\; p\_2\; p\_3\; (\; r\_1\; r\_2+\; r\_1\; r\_3\; +\; r\_2\; r\_3)\; +\; r\_1\; r\_2\; r\_3\; (p\_1\; p\_2\; +\; p\_1\; p\_3\; +\; p\_2\; p\_3)\; -\; 2\; p\_1\; p\_2\; p\_3\; r\_1\; r\_2\; r\_3\; =\; 0$

Inhomogeneous martini-A (3-7) lattice

Inhomogeneous martini-B (3-5) lattice

Inhomogeneous checkerboard lattice (conjecture) [cite journal
last = Wu
first = F. Y.
authorlink =
coauthors =
title = Critical point of planar Potts models
journal = Journal of Physics C
volume = 12
issue =
year = 1979
pages = L645–L650
doi = 10.1088/0022-3719/12/17/002]

$1\; -\; (p\_1\; p\_2\; +\; p\_1\; p\_3\; +\; p\_1\; p\_4\; +\; p\_2\; p\_3\; +\; p\_2\; p\_4\; +\; p\_3\; p\_4)\; +\; p\_1\; p\_2\; p\_3\; +\; p\_1\; p\_2\; p\_4\; +\; p\_1\; p\_3\; p\_4\; +\; p\_2\; p\_3\; p\_4\; =\; 0$

ee also

* Percolation
* Percolation theory
* 2D percolation cluster
* Effective Medium Approximations

References