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

TL;DR

This paper extends the understanding of the Random Cluster Model's connectivity functions and pressure from cubic lattices to a broader class of infinite graphs, demonstrating analyticity and decay properties in various regimes.

Contribution

It generalizes previous results on the Random Cluster Model to wider classes of infinite graphs, including those with minimal cut-set properties and isoperimetric inequalities.

Findings

Connectivity functions are analytic and decay exponentially in subcritical phase.

Finite connectivity functions are analytic in supercritical phase for certain graphs.

Decay of finite connectivity can be polynomial depending on graph topology.

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.