Uniqueness of maximal entropy measure on essential spanning forests

Abstract

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 $\mathrm{Aut}(G)$-invariant measure ${\mu }_G$ on essential spanning forests of $G$. We show that if $G$ is a connected, amenable graph and $\Gamma \subset \mathrm{Aut}(G)$ 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.

Published 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)