Uniqueness of maximal entropy measure on essential spanning forests
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Scott Sheffield [1]
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[1] Microsoft Research and University of California, Berkeley
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 34, Nº. 3, 2006, págs. 857-864
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
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.
