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Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs

Author: Tom Hutchcroft
Journal: J. Amer. Math. Soc. 33 (2020), 1101-1165
MSC (2010): Primary 60B99, 60K35
Published electronically: September 23, 2020
MathSciNet review: 4155221
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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 nonempty phase in which there are infinite light clusters, which implies the existence of a nonempty phase in which there are infinitely many infinite clusters. That is, we show that $p_c<p_h \leq p_u$ for any such graph. This answers a question of Häggström, 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 $T_k\times \mathbb {Z}^d$ of a $k$-regular tree with $\mathbb {Z}^d$ for $k\geq 3$ and $d \geq 1$, 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 $\mathbb {Z}^d$ edges are given different retention probabilities. These features had only previously been established for $d=1$, $k$ large.

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Additional Information

Tom Hutchcroft
Affiliation: Statslab, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambride, United Kingdom
MR Author ID: 1126713

Received by editor(s): November 29, 2017
Received by editor(s) in revised form: July 9, 2019, and February 23, 2020
Published electronically: September 23, 2020
Additional Notes: This work mainly took place while the author was a Ph.D. student at the University of British Columbia, during which time he was supported by a Microsoft Research Ph.D. Fellowship.
Article copyright: © Copyright 2020 American Mathematical Society