Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs
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 < 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 x Z^d of a k-regular tree with Z^d 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 Z^d edges are given different retention probabilities. These features had only previously been established for d = 1, k large.
Additional Information© 2020 American Mathematical Society. Received by the editors November 29, 2017, and, in revised form, July 9, 2019, and February 23, 2020. 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. The author thanks Nicolas Curien, Hugo Duminil-Copin, Matan Harel, Aran Raoufi, and Yinon Spinka for useful discussions, particularly during a visit by the author to the IHES in March 2017. He also thanks Omer Angel, Gady Kozma, Russ Lyons, and Asaf Nachmias for several helpful discussions, and also Russ Lyons for catching some typos in a previous version. Finally, we thank the three anonymous referees and Pengfei Tang for their close reading and detailed comments, which have greatly improved the paper.
Accepted Version - 1711.02590.pdf