Schramm's locality conjecture for critical probabilities

Let $(G_n)_{n\geq 1}$ be a sequence of transitive graphs converging locally to a transitive graph $G$. For a transitive graph $G$, let $p_c(G)$ denote the critical probability for Bernoulli bond percolation, and suppose that

$$\varlimsup_{n\to\infty} p_c(G_n)<1.$$

Schramm's locality conjecture. Then

$$p_c(G_n)\to p_c(G)\quad\text{as }n\to\infty.$$

This conjecture asserts that, provided the critical probabilities stay uniformly away from $1$ in the limsup, the critical probability is determined by local graph geometry. The paper verifies it under a uniform exponential lower bound on volume growth, including the case of uniformly nonamenable graphs; the general statement is the principal subject of the paper.