Percolation on finite graphs and isoperimetric inequalities

Consider a uniform expanders family G_n with a uniform bound on the degrees. It is shown that for any p and c > 0, a random subgraph of G_n obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|G_n|, with probability going to one, uniformly in p. The method from Ajtai, Komlós and Szemerédi [Combinatorica 2 (1982) 1-7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented.