Bond percolation

In statistical physics and mathematics, percolation theory describes the behaviour of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article percolation.

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 $n \times n \times n$ 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? Similarly, one can ask, given a connected graph at what fraction $1 - p$ of failures the graph will become disconnected (no large component).

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