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November 2019 Strict monotonicity of percolation thresholds under covering maps
Sébastien Martineau, Franco Severo
Ann. Probab. 47(6): 4116-4136 (November 2019). DOI: 10.1214/19-AOP1355

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 $\mathcal{G}=(V,E)$ be a quasi-transitive graph with $p_{c}(\mathcal{G})<1$, and let $G$ be a nontrivial group that acts freely on $V$ by graph automorphisms. Assume that $\mathcal{H}:=\mathcal{G}/G$ is quasi-transitive. Then one has $p_{c}(\mathcal{G})<p_{c}(\mathcal{H})$.

We provide results beyond this setting: we treat the case of general covering maps and provide a similar result for the uniqueness parameter $p_{u}$, under an additional assumption of boundedness of the fibres. The proof makes use of a coupling built by lifting the exploration of the cluster, and an exploratory counterpart of Aizenman–Grimmett’s essential enhancements.

Citation

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Sébastien Martineau. Franco Severo. "Strict monotonicity of percolation thresholds under covering maps." Ann. Probab. 47 (6) 4116 - 4136, November 2019. https://doi.org/10.1214/19-AOP1355

Information

Received: 1 September 2018; Revised: 1 January 2019; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212179
MathSciNet: MR4038050
Digital Object Identifier: 10.1214/19-AOP1355

Subjects:
Primary: 60K35, 82B43
Secondary: 82B26

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • November 2019
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