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Uniqueness of Maximal Entropy Measure on Essential Spanning Forests

Scott Sheffield
The Annals of Probability
Vol. 34, No. 3 (May, 2006), pp. 857-864
Stable URL: http://www.jstor.org/stable/25449893
Page Count: 8
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Uniqueness of Maximal Entropy Measure on Essential Spanning Forests
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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 Aut(G)-invariant measure $\mu _{G}$ on essential spanning forests of G. We show that if G is a connected, amenable graph and $\Gamma \subset {\rm Aut}(G)$ acts quasitransitively on G, then $\mu _{G}$ 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 $\Gamma \cong {\Bbb Z}^{d}$. Lyons discovered the error and asked about the more general statement that we prove.

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