## Uniqueness of maximal entropy measure on essential spanning forests.(English)Zbl 1106.60012

Summary: An essential spanning forest of an infinite graph $$G$$ is a spanning forest of $$G$$ in which all trees have infinitely many vertices. Let $$G_n$$ be an increasing sequence of finite connected subgraphs of $$G$$ for which $$\bigcup G_n=G$$. Pemantle’s arguments imply that the uniform measures on spanning trees of $$G_n$$ converge weakly to an $$\operatorname {Aut}(G)$$-invariant measure $$\mu_G$$ on essential spanning forests of $$G$$. We show that if $$G$$ is a connected, amenable graph and $$\Gamma \subset \operatorname {Aut}(G)$$ acts quasitransitively on $$G$$, then $$\mu_G$$ is the unique $$\Gamma$$-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 $$\Gamma\cong\mathbb Z^d$$. Lyons discovered the error and asked about the more general statement that we prove.

### MSC:

 60D05 Geometric probability and stochastic geometry

### Keywords:

amenable; ergodic; specific entropy
Full Text:

### References:

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