Strict monotonicity of percolation thresholds under covering maps

2018/03/26 by Martineau, Sébastien, Severo, Franco · 2 citations
#60K35 #82B43 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.1803.09686

Abstract

We answer a question of Benjamini and Schramm by proving that under reasonable conditions, quotienting a graph strictly increases the value of its percolation critical parameter p_c. More precisely, let G=(V,E) be a quasi-transitive graph with p_c(G)<1, and let G be a nontrivial group that acts freely on V by graph automorphisms. Assume that H:=G/G is quasi-transitive. Then one has p_c(G)

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• Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry
• Strict monotonicity of critical points in independent long-range percolation models

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