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Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs

Accepted version
Peer-reviewed

Type

Article

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Authors

Hutchcroft, Tom 

Abstract

We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a non-empty phase in which there are infinite light clusters, which implies the existence of a non-empty phase in which there are infinitely many infinite clusters. That is, we show that pc<phpu for any such graph. This answers a question of Haggstrom, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which implies that various critical exponents exist and take their mean-field values. All our results apply, for example, to the product Tk×Zd of a k-regular tree with Zd for k≥3 and d≥1, for which these results were previously known only for large k. Furthermore, our methods also enable us to establish the basic topological features of the phase diagram for anisotropic percolation on such products, in which tree edges and Zd edges are given different retention probabilities. These features had only previously been established for d=1, k large.

Description

Keywords

math.PR, math.PR, math-ph, math.MP

Journal Title

Journal of the American Mathematical Society

Conference Name

Journal ISSN

0894-0347
1088-6834

Volume Title

33

Publisher

American Mathematical Society (AMS)

Rights

All rights reserved