TABLE OF CONTENTS We now consider the random grid network with edge probability p (bond percolation). We define a connected component as a maximal set of vertices and edges such that for any two vertices x, y in the set, there exists an alternating sequence of distinct vertices and edges that starts with x and ends with y. In other words, x and y are in the same component if we can walk from one to the other over edges that are present. All the results we present also hold in the case of a random grid where each site is occupied independently with probability p (site percolation) and at the end of this section we shall see that very little is needed to accommodate the proofs. When the parameter is p, we write P p for the probability measure involved.
A phase transition in the random grid occurs at a critical value 0 < p c < 1. Namely, when p exceeds p c the random grid network contains a connected subgraph formed by an unbounded collection of vertices with probability one or, otherwise stated, almost surely (a.s.). In this case we say that the network percolates, or equivalently that the percolation model is supercritical. Conversely, when p < p c the random grid is a.s. composed of connected components of finite size, and we say that the model...
