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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
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References:

[1] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1–65. · Zbl 1016.60009
[2] Burton, R. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505. · Zbl 0662.60113
[3] Burton, R. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21 1329–1371. · Zbl 0785.60007
[4] Kenyon, R., Okounkov, A. and Sheffield, S. (2006). Dimers and amoebas. Ann. Math. · Zbl 1154.82007
[5] Lindenstrauss, E. (1999). Pointwise theorems for amenable groups. Electron. Res. Announc. Amer. Math. Soc. 5 82–80. · Zbl 0944.28014
[6] Lyons, R. (2005). Asymptotic enumeration of spanning trees. Combin. Probab. Comput. 14 491–522. · Zbl 1076.05007
[7] Ornstein, D. and Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math. 48 1–142. · Zbl 0637.28015
[8] Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559–1574. · Zbl 0758.60010
[9] Sheffield, S. (2003). Random surfaces: Large deviations and Gibbs measure classifications. Ph.D. dissertation, Stanford.
[10] Wilson, D. (1996). Generating random spanning trees more quickly than the cover time. Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing 296–303. ACM, New York. · Zbl 0946.60070
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