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.
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