Sharpness and locality for percolation on finite transitive graphs

Let $(G_n) = \left((V_n,E_n)\right)$ be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that $\lvert V_n \rvert \to \infty$ as $n \to \infty$. We say that percolation on $G_n$ has a sharp phase transition (as $n \to \infty$) if, as the percolation parameter crosses some critical point, the number of vertices contained in the largest percolation cluster jumps from logarithmic to linear order with high probability. We prove that percolation on $G_n$ has a sharp phase transition unless, after passing to a subsequence, the rescaled graph-metric on $G_n$ (rapidly) converges to the unit circle with respect to the Gromov-Hausdorff metric. We deduce that under the same hypothesis, the critical point for the emergence of a giant (i.e. linear-sized) cluster in $G_n$ coincides with the critical point for the emergence of an infinite cluster in the Benjamini-Schramm limit of $(G_n)$, when this limit exists.