Percolation pp 1-31 | Cite as

# 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 ℤ^{2} be the plane square lattice and let *p* be a number satisfying 0 ≤ *p* ≤ 1. We examine each edge of ℤ^{2} in turn, and declare this edge to be *open* with probability *p* and *closed* otherwise, independently of all other edges. The edges of ℤ^{2} 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 ℤ^{2} (see Figure 1.1), perhaps those vertices and edges of ℤ^{2} contained in some specified connected subgraph of ℤ^{2}. On immersion of the stone in water, a vertex *x* inside the stone is wetted if and only if there exists a path in ℤ^{2} 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’.

## Keywords

Ising Model Open Vertex Critical Phenomenon Open Cluster Percolation Theory## Preview

Unable to display preview. Download preview PDF.