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Around 2008, Schramm conjectured that the critical probabilities forBernoulli bond percolation satisfy the following continuity property: If$(G_n)_{n\geq 1}$ is a sequence of transitive graphs converging locally to atransitive graph $G$ and $\limsup_{n\to\infty} p_c(G_n) < 1$, then $p_c(G_n)\to p_c(G)$ as $n\to\infty$. We verify this conjecture under theadditional hypothesis that there is a uniform exponential lower bound on thevolume growth of the graphs in question. The result is new even in the casethat the sequence of graphs is uniformly nonamenable.  In the unimodular case, this result is obtained as a corollary to thefollowing theorem of independent interest: For every $g>1$ and $M<\infty$,there exist positive constants $C=C(g,M)$ and $\delta=\delta(g,M)$ such that if$G$ is a transitive unimodular graph with degree at most $M$ and growth$\operatorname{gr}(G) := \inf_{r\geq 1} |B(o,r)|^{1/r}\geq g$, then \[\mathbf{P}_{p_c} \bigl(|K_o|\geq n\bigr) \leq C n^{-\delta} \] for every $n\geq1$, where $K_o$ is the cluster of the root vertex $o$. The proof of thisinequality makes use of new universal bounds on the probabilities of certaintwo-arm events, which hold for every unimodular transitive graph.

Locality of the critical probability for transitive graphs of exponential growth