Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry

We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying $p_c(\mathscr{G})<1$ and take $p>p_c(\mathscr{G})$. Let $\mathscr{H}$ be another such graph and assume that $\mathscr{G}$ and $\mathscr{H}$ have the same ball of radius $r$ for $r$ large. We prove that various quantities regarding percolation of parameter close to $p$ on $\mathscr{H}$ can be well understood from $(\mathscr{G},p)$ alone. This includes uniform versions of supercritical sharpness as well as the Kesten-Zhang bound on the probability of observing a large finite cluster: the constants involved can be chosen to depend only on $(\mathscr{G},p)$. We also prove that $\theta_\mathscr{H}$ is an analytic function of $p$ in the whole supercritical regime and that, for a suitable $\varepsilon=\varepsilon(\mathscr{G},p)>0$, the analytic extension of $\theta_\mathscr{H}$ to the $\varepsilon$-neighbourhood of $p$ in $\mathbb C$ is, uniformly, well approximated by the analytic extension of $\theta_\mathscr{G}$. The proof relies on new results on the connectivity of minimal cutsets; in particular, we answer a question asked by Babson and Benjamini in 1999. We further discuss connections with the conjecture of non-percolation at criticality.