Uniqueness of maximal entropy measure on essential spanning forests
نویسنده
چکیده
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 (1991) 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 quasi-transitively 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 (1993), who gave a short but incorrect proof in the case Γ ∼= Z. Lyons discovered the error (2002) and asked about the more general statement that we prove.
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