un 2 00 1 percolation on finite graphs Itai Benjamini

Several questions and few answers regarding percolation on finite graphs are presented. The following is a note regarding the asymptotic study of percolation on finite transitive graphs. On the one hand, the theory of percolation on infinite graphs is rather developed, although still with many open problems (See [9]). On the other hand random graphs were deeply studied (see [7]). Finite transitive graphs are somewhat in the middle. Though a big part of the picture is similar to the percolation on infinite graphs theory, some qualitative phenomena are different and provides new challenges. Recall that a graph G is transitive iff for every two vertices v, u ∈ G, there is an isomorphism of G, mapping v to u. See [4] or [9] for further background and definitions. Given a finite graph G, delete edges from G independently, each with probability 1 − p. Denote by p G = p G 1/2 , the threshold probability for having a connected component of size |G|/2. I.e., P p G There is a connected component of size ≥ |G|/2 = 1/2. Note that for transitive graphs the threshold for having a large connected component is sharp [5]. Write p G α for the threshold probability for having a component of size α|G|, with probability 1/2.