What is . . .Percolation?

WHAT IS...?Percolation?Harry Kesten2Percolation is a simple probabilistic model which v∈Z as the collection of points connected to vexhibits a phase transition (as we explain below). by an open path. The clusters C(v) are the maxi-2The simplest version takes place on Z , which we mal connected components of the collection of2view as a graph with edges between neighboring open edges of Z , and θ(p) is the probability that2vertices. All edges of Z are, independently of eachC(0) is infinite. If pp, then θ(p)> 0 and there is acists an open path, i.e., a path all of whose edges are strictly positive probability that C(0) is infinite.open, from the origin to the exterior of the square An application of Kolmogorov’s zero-one law shows2S := [−n,n] ?” This question was raised by Broad-n that there is then with probability 1 some infinitebent in 1954 at a symposium on Monte Carlo meth- cluster. In fact, it turns out that there is a uniqueods. It was then taken up by Broadbent and Ham- infinite cluster. Thus, the global behavior of the sys-mersley, who regarded percolation as a model for tem is quite different for p

p.c ca random medium. They interpreted the edges of ...