Non-triviality of the phase transition for percolation on finite transitive graphs

We prove that if (G₍) ₍ ₁= ( (V₍, E₍) ) ₍ ₁ is a sequence of finite, vertex-transitive graphs with bounded degrees and |V₍| that is at least (1+) -dimensional for some >0 in the sense that diam (G₍) =O (|V₍|^1/ (1+) ) as n then this sequence of graphs has a non-trivial phase transition for Bernoulli bond percolation. More precisely, we prove under these conditions that for each 00 such that every infinite graph G of degree at most k whose vertex set has at most n orbits under Aut (G) has either pc=1 or pc 1-.