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May 2006 Uniqueness of maximal entropy measure on essential spanning forests
Scott Sheffield
Ann. Probab. 34(3): 857-864 (May 2006). DOI: 10.1214/009117905000000765


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.


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Scott Sheffield. "Uniqueness of maximal entropy measure on essential spanning forests." Ann. Probab. 34 (3) 857 - 864, May 2006.


Published: May 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1106.60012
MathSciNet: MR2243870
Digital Object Identifier: 10.1214/009117905000000765

Primary: 60D05

Keywords: Amenable , Ergodic , essential spanning forest , Specific entropy

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 3 • May 2006
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