Locality of the critical probability for transitive graphs of exponential growth

Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_n)_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive 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 the additional hypothesis that there is a uniform exponential lower bound on the volume growth of the graphs in question. The result is new even in the case that the sequence of graphs is uniformly nonamenable. In the unimodular case, this result is obtained as a corollary to the following 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 $n\geq 1$, where $K_o$ is the cluster of the root vertex $o$. The proof of this inequality makes use of new universal bounds on the probabilities of certain two-arm events, which hold for every unimodular transitive graph.