Uniqueness of maximal entropy measure on essential spanning forests

• 成果类型: Article
• 署名作者: Sheffield, Scott
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000765
• 发表日期: 2006
• 页码: 857-864
• 关键词: amenable-groups; THEOREMS; trees

摘要

An essential spanning forest of an infinite graph G is a spanning forest of G in which all trees have infinitely many vertices. Let G(n) be an increasing sequence of finite connected subgraphs of G for which boolean OR G(n) = G. Pemantle's arguments imply that the uniform measures on spanning trees of G, converge weakly to an Aut(G)-invariantmeasure mu(G) on essential spanning forests of G. We show that if G is a connected, amenable graph and F subset of Aut(G) acts quasitransitively on G, then mu(G) is the unique Gamma-invariant measure on essential spanning forests of G for which the specific entropy is maximal. This result originated with Burton and Pemantle, who gave a short but incorrect proof in the case Gamma congruent to Z(d). Lyons discovered the error and asked about the more general statement that we prove.