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Published May 2020 | Accepted Version
Journal Article Open

Locality of the critical probability for transitive graphs of exponential growth


Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If (G_n)_(n ≥ 1) is a sequence of transitive graphs converging locally to a transitive graph G and limsup_(n → ∞)p_c(G_n) < 1, then p_c(G_n) → p_c(G) as n → ∞. We verify this conjecture under the additional hypothesis that there is a uniform exponential lower bound on the volume growth of the graphs in question. The result is new even in the case that the sequence of graphs is uniformly nonamenable. In the unimodular case, this result is obtained as a corollary to the following theorem of independent interest: For every g > 1 and M < ∞, there exist positive constants C = C(g,M) and δ = δ(g,M) such that if G is a transitive unimodular graph with degree at most M and growth gr(G):= inf_(r ≥ 1)|B(o,r)|^(1/r) ≥ g, then P_(p_c)(|K_o| ≥ n) ≤ C_n^(−δ) for every n ≥ 1, where K_o is the cluster of the root vertex o. The proof of this inequality makes use of new universal bounds on the probabilities of certain two-arm events, which hold for every unimodular transitive graph.

Additional Information

© 2020 Institute of Mathematical Statistics. Received: 1 August 2018; Revised: 1 April 2019; Published: May 2020. First available in Project Euclid: 17 June 2020. We thank Jonathan Hermon for many helpful discussions, and for his careful reading of an earlier version of this manuscript. This paper also greatly benefited from discussions with Vincent Tassion on rewriting the Aizenman-Kesten-Newman uniqueness proof with martingale techniques, which took place at the Isaac Newton Institute during the RGM follow up workshop. We also thank Itai Benjamini, Hugo Duminil-Copin, Gady Kozma, Russ Lyons, Sebastien Martineau, Asaf Nachmias, and an anonymous referee for comments on earlier versions of the paper.

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