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

TL;DR

This work proves that in transitive graphs of polynomial growth, key percolation observables in the supercritical regime exhibit locality: knowing a large finite ball of one graph suffices to understand observables on any nearby graph. It introduces interfaces and a coarse connectivity framework for minimal cutsets, builds a renormalised analytic representation of the percolation function $ heta_{\mathscr{G}}(p)$, and establishes that $ heta_{\mathscr{G}}(p)$ is analytic on $(p_c(\mathscr{G}),1]$ with uniform holomorphic extensions across local graph limits. The authors also prove uniform supercritical sharpness and control the probability of large finite clusters, along with a local quasi-isometry invariance of cutconnectivity constants, answering a longstanding question of Babson and Benjamini. Collectively, these results advance a local-to-global understanding of percolation beyond $p_c$, with implications for the regularity of $ heta$ and the relationship between criticality and locality across graph families.

Abstract

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 $θ_\mathscr{H}$ 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 $θ_\mathscr{H}$ to the $\varepsilon$-neighbourhood of $p$ in $\mathbb C$ is, uniformly, well approximated by the analytic extension of $θ_\mathscr{G}$. 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.

Figures (4)

• Figure 1: Illustration of $\tilde{\Pi}$.
• Figure 2: In the top left corner, we depict the vertices $x$ and $y$ in the graph $\mathscr{H}$ as crosses, and the vertices of $\Pi$ as dots. In the top right corner, we depict the image under $\psi$, where we have highlighted the ${C"}$-neighbourhood of $\psi(\Pi)$ in red. In the bottom right corner, we depict a minimal cutset between $\psi(x)$ and $\psi(y)$ in the ${C"}$-neighbourhood of $\psi(\Pi)$, and in the bottom left corner, its image under $\varphi$. The latter gives no information on the connectivity of $\Pi$.
• Figure 3: An illustration of the choice of the minimal cutset in the proof of Proposition \ref{['prop:qi']}. In order to keep both parts, we do try to separate $\psi(x)$ from $\psi(y)$ but from $v$.
• Figure 4: An example where we need to replace both vertices $\psi(x)$ and $\psi(y)$, and instead consider a minimal cutset with respect to two new vertices $u$ and $v$.

Theorems & Definitions (61)

• Theorem 1.1: Contreras--Martineau--Tassion, 2024
• Theorem 1.2
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8
• Proposition 1.9