Uniqueness of maximal entropy measure on essential spanning forests

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dc.creatorSheffield, Scott
dc.date2004-06-25
dc.date2006-06-29
dc.date.accessioned2026-07-07T06:36:53Z
dc.date.available2026-07-07T06:36:53Z
dc.descriptionAn 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 $μ_G$ on essential spanning forests of $G$. We show that if $G$ is a connected, amenable graph and $Γ\subset \operatorname {Aut}(G)$ acts quasitransitively on $G$, then $μ_G$ is the unique $Γ$-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 $Γ\cong\mathbb{Z}^d$. Lyons discovered the error and asked about the more general statement that we prove.
dc.descriptionPublished at http://dx.doi.org/10.1214/009117905000000765 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/math/0406513
dc.identifierhttp://arxiv.org/abs/math/0406513
dc.identifierAnnals of Probability 2006, Vol. 34, No. 3, 857-864
dc.identifierdoi:10.1214/009117905000000765
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/100235
dc.subjectProbability
dc.subjectCombinatorics
dc.subject60D05 (Primary)
dc.titleUniqueness of maximal entropy measure on essential spanning forests
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