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 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.

Procacci, A., Scoppola, B. (2008). Convergent expansions for Random Cluster Model with q > 0 on infinite graphs. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 7(5), 1145-1178 [10.3934/cpaa.2008.7.1145].

Convergent expansions for Random Cluster Model with q > 0 on infinite graphs

SCOPPOLA, BENEDETTO
2008-01-01

Abstract

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 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.
2008
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/07 - FISICA MATEMATICA
English
Con Impact Factor ISI
cluster expansion; Random Cluster Model; infinite graphs
Procacci, A., Scoppola, B. (2008). Convergent expansions for Random Cluster Model with q > 0 on infinite graphs. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 7(5), 1145-1178 [10.3934/cpaa.2008.7.1145].
Procacci, A; Scoppola, B
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/32303
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