Gap at 1 for the percolation threshold of Cayley graphs

TL;DR

This paper proves a gap at 1 in the possible percolation thresholds of Cayley graphs, showing thresholds are either exactly 1 or bounded away from 1 by a positive constant.

Contribution

It introduces a novel approach combining Gaussian free field techniques with group structure theorems to establish the gap at 1 for Cayley graph percolation thresholds.

Findings

Percolation thresholds of Cayley graphs are either 1 or at most 1 - ε₀.

Established a uniform gap at 1 for all Cayley graphs.

Connected percolation phase transition properties with group structure.

Abstract

We prove that the set of possible values for the percolation threshold $p_c$ of Cayley graphs has a gap at 1 in the sense that there exists $\varepsilon_0>0$ such that for every Cayley graph $G$ one either has $p_c(G)=1$ or $p_c(G) \leq 1-\varepsilon_0$. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo & Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green & Tao.