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

2025/11/03 by Sébastien Martineau, Christoforos Panagiotis, Martineau, Sébastien +1
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2511.01851

openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

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 \mathscrG satisfying p_c(\mathscrG)<1 and take p>p_c(\mathscrG). Let \mathscrH be another such graph and assume that \mathscrG and \mathscrH have the same ball of radius r for r large. We prove that various quantities regarding percolation of parameter close to p on \mathscrH can be well understood from (\mathscrG,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 (\mathscrG,p). We also prove that θ_\mathscrH is an analytic function of p in the whole supercritical regime and that, for a suitable ε=ε(\mathscrG,p)>0, the analytic extension of θ_\mathscrH to the ε-neighbourhood of p in \mathbb C is, uniformly, well approximated by the analytic extension of θ_\mathscrG. 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.

Citations

• Counting minimal cutsets and p_c<1
• Sharpness and locality for percolation on finite transitive graphs
• Balls in groups: volume, structure and growth
• Locality of percolation for graphs with polynomial growth
• Supercritical percolation on graphs of polynomial growth
• Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs
• A finitary structure theorem for vertex-transitive graphs of polynomial growth
• No percolation at criticality on certain groups of intermediate growth
• Analyticity results in Bernoulli Percolation
• Locality of the critical probability for transitive graphs of\n exponential growth
• Strict monotonicity of percolation thresholds under covering maps
• Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs
• Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters
• Mean-field behavior for nearest-neighbor percolation in d&gt;10
• Absence of infinite cluster for critical Bernoulli percolation on slabs
• Locality of percolation for abelian Cayley graphs
• The structure of approximate groups
• A finitary version of Gromov's polynomial growth theorem
• Is the critical percolation probability local?

Related

• Supercritical percolation on graphs of polynomial growth
• Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution
• Locality of percolation for graphs with polynomial growth
• From local giants to locality in long-range percolation
• The polynomial growth of the infinite long-range percolation cluster