Spread-out percolation on transitive graphs of polynomial growth

TL;DR

This work establishes that for any vertex-transitive graph $G$ with superlinear polynomial growth, the spread-out percolation threshold on the scale-$r$ graph $G_r$ satisfies $p_c(G_r)=(1+o(1))/\beta_G(r)$ as $r\to\infty$, equivalently $p_c(G_r)\sim 1/\deg(G_r)$. The authors develop a multi-layered approach: (i) reduce to Cayley graphs of torsion-free nilpotent groups, (ii) pass to the asymptotic cone and describe local percolation via a limit kernel, (iii) treat the local model as an inhomogeneous random graph with a converging kernel, and (iv) apply a renormalization argument together with a $K$-independent percolation framework to obtain an infinite cluster. A key technical contribution is a Haar-measure convergence result for rescaled lattices on the asymptotic cone, which then feeds the inhomogeneous random-graph analysis. The results verify a special case of a conjecture by Easo and Hutchcroft and connect percolation thresholds to a group-theoretic dimension notion introduced by Georgakopoulos, highlighting a robust methodology for analyzing random processes on polynomial-growth graphs. The methods blend geometric group theory (Malcev closures, Pansu limits), probabilistic kernel methods (inhomogeneous random graphs), and renormalization to extend Penrose-type percolation phenomena beyond $\mathbb{Z}^d$.

Abstract

Let $G$ be a vertex-transitive graph of superlinear polynomial growth. Given $r>0$, let $G_r$ be the graph on the same vertex set as $G$, with two vertices joined by an edge if and only if they are at graph distance at most $r$ apart in $G$. We show that the critical probability $p_c(G_r)$ for Bernoulli bond percolation on $G_r$ satisfies $p_c(G_r) \sim 1/\mathrm{deg}(G_r)$ as $r\to\infty$. This extends work of Penrose and Bollobás-Janson-Riordan, who considered the case $G=\mathbb{Z}^d$. Our result provides an important ingredient in parallel work of Georgakopoulos in which he introduces a new notion of dimension in groups. It also verifies a special case of a conjecture of Easo and Hutchcroft.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark
• Remark
• Remark
• Proposition 3.1
• Proposition 3.2