Percolation theory

Introduction

A representative question (and the source of the name) is as follows. Assume that some liquid is poured on top of some porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is modelled mathematically as a three-dimensional network of n × n × n vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability p, or closed with probability 1 – p, and they are assumed to be independent. Therefore, for a given p, what is the probability that an open path (meaning a path, each of whose links is an "open" bond) exists from the top to the bottom? The behavior for large n is of primary interest. This problem, called now bond percolation, was introduced in the mathematics literature by Broadbent & Hammersley (1957), and has been studied intensively by mathematicians and physicists since then.

In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability p or "empty" (in which case its edges are removed) with probability 1-p; the corresponding problem is called site percolation. The question is the same: for a given p, what is the probability that a path exists between top and bottom?

Of course the same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By Kolmogorov's zero–one law, for any given p, the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of p (proof via coupling argument), there must be a critical p (denoted by p_c) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for n as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of p.

In some cases p_c may be calculated explicitly. For example, for the square lattice Z² in two dimensions, p_c = 1/2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, see Kesten (1982). A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at p_c = 1/(z − 1) for a coordination number z. For most infinite lattice graphs, p_c cannot be calculated exactly.

Universality

The universality principle states that the numerical value of p_c is determined by the local structure of the graph, whereas the kind of behavior of clusters that is observed below, at, and above p_c is independent of the local structure, and therefore, in some sense these behaviors are more natural to consider than p_c itself . This universality also means that for a given dimension, the various critical exponents, the fractal dimension of the clusters at p_c is independent of the lattice type and percolation type (e.g., bond or site). However, recently percolation has been performed on a weighted planar stochastic lattice (WPSL) and found that albeit the dimension of the WPSL coincides with the dimension of the space where it is embedded its universality class is different from that of all the known planar lattices.

Subcritical and supercritical

The main fact in the subcritical phase is "exponential decay". That is, when p < p_c, the probability that a specific point (for example, the origin) is contained in an open cluster (meaning a maximal connected set of "open" edges of the graph) of size r decays to zero exponentially in r. This was proved for percolation in three and more dimensions by Menshikov (1986) and independently by Aizenman & Barsky (1987). In two dimensions, it formed part of Kesten's proof that p_c = 1/2.

The dual graph of the square lattice Z² is also the square lattice. It follows that, in two dimensions, the supercritical phase is dual to a subcritical percolation process. This provides essentially full information about the supercritical model with d = 2. The main result for the supercritical phase in three and more dimensions is that, for sufficiently large N, there is an infinite open cluster in the two-dimensional slab Z² × [0, N]^d−2. This was proved by Grimmett & Marstrand (1990).

In two dimensions with p < 1/2, there is with probability one a unique infinite closed cluster (a closed cluster is a maximal connected set of "closed" edges of the graph) . Thus the subcritical phase may be described as finite open islands in an infinite closed ocean. When p > 1/2 just the opposite occurs, with finite closed islands in an infinite open ocean. The picture is more complicated when d ≥ 3 since p_c < 1/2, and there is coexistence of infinite open and closed clusters for p between p_c and 1 − p_c.

Critical

The model has a singularity at the critical point p = p_c believed to be of power-law type. Scaling theory predicts the existence of critical exponents, depending on the number d of dimensions, that determine the class of the singularity. When d = 2 these predictions are backed up by arguments from conformal field theory and Schramm–Loewner evolution, and include predicted numerical values for the exponents. Most of these predictions are conjectural except when the number d of dimensions satisfies either d = 2 or d ≥ 19. They include:

See Grimmett (1999). In dimension ≥ 19, these facts are largely proved using a technique known as the lace expansion. It is believed that a version of the lace expansion should be valid for 7 or more dimensions, perhaps with implications also for the threshold case of 6 dimensions. The connection of percolation to the lace expansion is found in Hara & Slade (1990).

In dimension 2, the first fact ("no percolation in the critical phase") is proved for many lattices, using duality. Substantial progress has been made on two-dimensional percolation through the conjecture of Oded Schramm that the scaling limit of a large cluster may be described in terms of a Schramm–Loewner evolution. This conjecture was proved by Smirnov (2001) in the special case of site percolation on the triangular lattice.

Different models