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

We prove that if (G_{n})_{n \geq1}=((V_{n},E_{n}))_{n\geq 1} is a sequence of finite, vertex-transitive graphs with bounded degrees and |V_{n}|\to\infty that is at least (1+\varepsilon) -dimensional for some \varepsilon>0 in the sense that \operatorname{diam} (G_{n})=O(|V_{n}|^{1/(1+\varepsilon)}) \quad \text{as }n\to\infty 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 0