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May 2020 Locality of the critical probability for transitive graphs of exponential growth
Tom Hutchcroft
Ann. Probab. 48(3): 1352-1371 (May 2020). DOI: 10.1214/19-AOP1395


Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_{n})_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$ and $\limsup_{n\to \infty }p_{c}(G_{n})<1$, then $p_{c}(G_{n})\to p_{c}(G)$ as $n\to \infty $. 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<\infty $, there exist positive constants $C=C(g,M)$ and $\delta =\delta (g,M)$ such that if $G$ is a transitive unimodular graph with degree at most $M$ and growth $\operatorname{gr}(G):=\inf_{r\geq 1}|B(o,r)|^{1/r}\geq g$, then \[\mathbf{P}_{p_{c}}\bigl(\vert K_{o}\vert \geq n\bigr)\leq Cn^{-\delta }\] for every $n\geq 1$, 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.


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Tom Hutchcroft. "Locality of the critical probability for transitive graphs of exponential growth." Ann. Probab. 48 (3) 1352 - 1371, May 2020.


Received: 1 August 2018; Revised: 1 April 2019; Published: May 2020
First available in Project Euclid: 17 June 2020

zbMATH: 07226363
MathSciNet: MR4112717
Digital Object Identifier: 10.1214/19-AOP1395

Primary: 60K35
Secondary: 82B43

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 3 • May 2020
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