The Annals of Probability

Uniqueness of maximal entropy measure on essential spanning forests

Scott Sheffield

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An essential spanning forest of an infinite graph G is a spanning forest of G in which all trees have infinitely many vertices. Let Gn be an increasing sequence of finite connected subgraphs of G for which Gn=G. Pemantle’s arguments imply that the uniform measures on spanning trees of Gn converge weakly to an Aut (G)-invariant measure μG on essential spanning forests of G. We show that if G is a connected, amenable graph and Γ⊂Aut (G) acts quasitransitively on G, then μG 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≅ℤd. Lyons discovered the error and asked about the more general statement that we prove.

Article information

Ann. Probab., Volume 34, Number 3 (2006), 857-864.

First available in Project Euclid: 27 June 2006

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Amenable essential spanning forest ergodic specific entropy


Sheffield, Scott. Uniqueness of maximal entropy measure on essential spanning forests. Ann. Probab. 34 (2006), no. 3, 857--864. doi:10.1214/009117905000000765.

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