Uniqueness of maximal entropy measure on essential spanning forests. (English) Zbl 1106.60012

Summary: 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 \(\bigcup G_n=G\). Pemantle’s arguments imply that the uniform measures on spanning trees of \(G_n\) converge weakly to an \(\operatorname {Aut}(G)\)-invariant measure \(\mu_G\) on essential spanning forests of \(G\). We show that if \(G\) is a connected, amenable graph and \(\Gamma \subset \operatorname {Aut}(G)\) 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 \(\Gamma\cong\mathbb Z^d\). Lyons discovered the error and asked about the more general statement that we prove.


60D05 Geometric probability and stochastic geometry
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