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 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.
Citation
Scott Sheffield. "Uniqueness of maximal entropy measure on essential spanning forests." Ann. Probab. 34 (3) 857 - 864, May 2006. https://doi.org/10.1214/009117905000000765
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