What is Percolation?

Abstract

Suppose we immerse a large porous stone in a bucket of water. What is the probability that the centre of the stone is wetted? In formulating a simple stochastic model for such a situation, Broadbent and Hammersley (1957) gave birth to the ‘percolation model’. In two dimensions their model amounts to the following. Let ℤ² be the plane square lattice and let p be a number satisfying 0 ≤ p ≤ 1. We examine each edge of ℤ² in turn, and declare this edge to be open with probability p and closed otherwise, independently of all other edges. The edges of ℤ² represent the inner passageways of the stone, and the parameter p is the proportion of passages which are broad enough to allow water to pass along them. We think of the stone as being modelled by a large, finite subsection of ℤ² (see Figure 1.1), perhaps those vertices and edges of ℤ² contained in some specified connected subgraph of ℤ². On immersion of the stone in water, a vertex x inside the stone is wetted if and only if there exists a path in ℤ² from x to some vertex on the boundary of the stone, using open edges only. Percolation theory is concerned primarily with the existence of such ‘open paths’.