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September  2008, 7(5): 1145-1178. doi: 10.3934/cpaa.2008.7.1145

Convergent expansions for random cluster model with $q>0$ on infinite graphs

A. Procacci 1, and  Benedetto Scoppola 2,

1. 

Department of Mathematics, Universidade Federal de Minas Gerais, 30161-970 Belo Horizonte
2. 

Dipartimento di Matematica Universitá “Tor Vergata” di Roma, V.le della ricerca scientifica, 00100 Roma, Italy

Received  August 2007 Revised  April 2008 Published  June 2008

In this paper we extend our previous results on the connectivity functions and pressure of the Random Cluster Model in the highly subcritical phase and in the highly supercritical phase, originally proved only on the cubic lattice $\mathbb Z^d$, to a much wider class of infinite graphs. In particular, concerning the subcritical regime, we show that the connectivity functions are analytic and decay exponentially in any bounded degree graph. In the supercritical phase, we are able to prove the analyticity of finite connectivity functions in a smaller class of graphs, namely, bounded degree graphs with the so called minimal cut-set property and satisfying a (very mild) isoperimetric inequality. On the other hand we show that the large distances decay of finite connectivity in the supercritical regime can be polynomially slow depending on the topological structure of the graph. Analogous analyticity results are obtained for the pressure of the Random Cluster Model on an infinite graph, but with the further assumptions of amenability and quasi-transitivity of the graph.
Keywords: Random Cluster Model, infinite graphs., Cluster expansion.
Mathematics Subject Classification: Primary: 82B20, 60K35; Secondary: 05C9.
Citation: A. Procacci, Benedetto Scoppola. Convergent expansions for random cluster model with $q>0$ on infinite graphs. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1145-1178. doi: 10.3934/cpaa.2008.7.1145
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