percolation theory

In mathematics, percolation theory describes the behavior of connected clusters in a random graph.

Introduction

A representatitve question (and the source of the name) is as follows. Assume we have some porous material and we pour some liquid on top. Will the liquid be able to make its way from hole to hole and reach the bottom? We model the physical question mathematically as a three dimensional network of

n\times n\times n

points (or vertices) the connections (or edges) between each two neighbors may be open (allowing the liquid through) with probability p, or closed with probability 1-p, and we assume they are independent. We ask: for a given p, what is the probability that an open path exists from the top to the bottom? Mostly we are interested in the behavior for large n. As is quite typical, it is actually easier to examine infinite network than just large ones. In this case the corresponding question is: does there exists an infinite open cluster? In this case we may use Kolmogorov's zero-one law to see that, for any given p, the probability that an infinite cluster exists is either zero or one. Since this probability is increasing (this is obvious intuitively, but mathematicians need a coupling argument to prove it), 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 increase sharply from very close to zero to very close to one in a short span of p-s. In some cases

p_c

may be calculated explicitly. For example, for the square lattice in two dimensions,

p_c

=½, a fact which was a open question for more than 20 years and was finally resolved by Harry Kesten in the early 80s. More often than not,

p_c

cannot be calculated. For example,

p_c

is not known in three dimensions. However, it turns out that calculating

p_c

is not necessarily the most interesting thing to do. The universality principle states that the value of

p_c

is connected to the local structure of the graph, while the behavior of clusters below, at and above

p_c

are invariants of the local structure, and therefore, in some sense are more natural quantities to consider. Sometimes it is easier to open and close vertices rather than edges. This is called site percolation while the model described above is more properly called bond percolation.

The subcritical and supercritical phases

For

p, the probability that a specific point (for example, zero) is contained in an open cluster of size

r drops exponentially in r. There is, with probability one, one infinite closed cluster. The finite closed clusters, like the open clusters, have exponential tails and thus are very small. Thus the subcritical phase may be described as open islands in a closed sea. The base of the exponent converges to one as p approaches

p_c

. When

p>p_c

just the opposite occurrs, with closed islands in an open sea. These facts were proved by Menshikov in 1986.

The critical phase

Arguments from quantum field theory and quantum gravitation make a sequence of impressive conjectures about the critical phase, most of which are unproved:

There are no infinite clusters (open or closed)
The probability that there is an open path from some fixed point (say zero) to a distance of r decreases polynomially, i.e. is on the order of $r^\alpha$ for some α
α does not depend on the particular lattice chosen, or on other local parameters. It depends only on the dimension (this is an instance of the universality principle).
$\alpha_d$ decreases from d = 2 until d = 6 and then stays fixed.
$\alpha_6=-1$
$\alpha_2=-5/48$ . The stochastic behavior of large clusters in two dimensions is conformally invariant.

In dimension ≥ 19, these facts (except universality) are proved. In dimension 2, the first fact ("no percolation in the critical phase") is proved for all latices, but the rest have only been proved for site percolation on the honeycomb or hexagonal lattice, and of course, universality is not proved. In dimensions 3 to 18, even the first conjecture is still open.

The different models

The first model studied was the Bernoulli percolation. In this models all bonds are independent.
A generealization next was introduced as the FK model, which has many connections with the Ising model and other Potts models.
A branch of the percolation theory study Bernoulli percolation on complete graphs.
Oriented percolation, which has connections with the contact process.
First passage percolation

Conjectures

In percolation questions are easy to ask, but hard to answer.

There is no percolation at the critical point in all dimensions

References

Percolation, by G.R. Grimmett
Percolation for mathematicians by H. Kesten

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