Percolation on finite cayley graphs

Abstract

In this paper, we study percolation on finite Cayley graphs. A conjecture of Benjamini says that the critical percolation pc of such a graph can be bounded away from one, for any Cayley graph satisfying a certain diameter condition. We prove Benjamini’s conjecture for some special classes of groups. We also establish a reduction theorem, which allows us to build Cayley graphs for large groups without increasing pc.