LOCALITY OF THE CRITICAL PROBABILITY FOR TRANSITIVE GRAPHS OF EXPONENTIAL GROWTH

• 成果类型: Article
• 署名作者: Hutchcroft, Tom
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1395
• 发表日期: 2020
• 页码: 1352-1371
• 关键词: critical percolation; infinite-cluster; PHASE-TRANSITION; uniqueness; sharpness; density

摘要

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 lim sup(n ->infinity) pc(G(n)) < 1, then p(c)(G(n)) -> p(c)(G) as n -> infinity. 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 < infinity, 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 gr(G) := inf(r >= 1) vertical bar B(o,r)vertical bar(1/r) >= g, then P-pc (vertical bar K-o vertical bar >= n) <= Cn(-delta) 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.