Gap at 1 for the percolation threshold of Cayley graphs

2021/10/31 by Christoforos Panagiotis, Panagiotis, Christoforos, Franco Severo +1
Mathematics · #60K35 #82B43 #FOS: Mathematics #Group Theory (math.GR) #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2111.00555

openalex publication_date 2021/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ε₀>0 such that for every Cayley graph G one either has p_c(G)=1 or p_c(G) ≤ 1-ε₀. 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.

Related

• Existence of a percolation threshold on finite transitive graphs
• Non-amenable Cayley graphs of high girth have p_c < p_u and mean-field exponents
• Haagerup property and group-invariant percolation
• Non-triviality of the phase transition for percolation on finite transitive graphs
• Critical threshold for regular graphs