What is Percolation?
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’.
KeywordsIsing Model Open Vertex Critical Phenomenon Open Cluster Percolation Theory
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