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

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. graphs with no circuits) , see e.g. [4,5,23,20,21,22,37,25,26,27,19,18]. There have been also a few papers dealing with percolation processes on quasi transitive or transitive graphs, including amenable graphs, see e.g.[2], [3], [30]. Roughly speaking, in a transitive graph G any vertex of the graph is equivalent; in other words G "looks the same" by observers sitting in different vertices. In a quasi-transitive graph G there is a finite number of different types of vertices and G "looks the same" by observers sitting in vertices of the same type.Some general results about percolation on general infinite graphs (i.e. not necessarily non amenable and/or quasi-transitive) appeared in [4], [5], and [2], and more recently in [33] (see also references therein). There are also some other works about the Potts model, in particular the antiferromagnetic case, on general finite graphs [38,39] and on amenable quasi-transitive infinite graphs [35]. In this paper we focus our attention on the study of the dependent percolation process known as Random Cluster Model (RCM) on general infinite graphs. The RCM was proposed by Fortuin and Kasteleyn in the early seventies [12] as a generalization of the Potts model. The RCM on a graph G depends on two real parameters: the parameter p ∈ [0, 1] and the parameter q ∈ (0, +∞). The parameter p represents the weight of an edge of G to be open independently of the other edges and it is related to the temperature of the Potts model. The parameter q, when different from 1, introduces a dependence in the percolation process described by RCM, and, when integer greater than 2, it represents the number of colors in the Potts model.Some results on RCM can be proved for all the values of the parameters q and p. In particular, there are results about the logarithm of the total weight of the measure (pressure). Namely, the existence of the pressure of the RCM, its independency on boundary conditions and its differentiability (with respect to p al...

Percolation on Infinite Graphs and Isoperimetric Inequalities

“…in the highly supercritical phase). In this case, however, they obtained (Theorem 5.9 of [19]) an upper bound for the connectivity functions of the form (C 1 , C 2 are constants)…”

“…In [19] the authors obtained (see there Theorem 4.1) an upper bound for the two-point connectivity function of the Random Cluster Model with parameters p ∈ [0, 1] and q > 0 on a bounded degree graph G, showing that this function decays at least exponentially in G as soon as, for fixed q, p is sufficiently close to zero (i.e. in the highly subcritical phase).…”

“…Now, it easy to provide examples of graphs for which f G (x, y)/d G (x, y) → 0 as d G (x, y) → ∞. So, being able get a lower bound of the same form of (3.3), one could raise the question (raised in fact in [19]) whether there are graphs for which the finite connectivity functions decay sub-exponentially, even for p arbitrarily close to 1. In this last section, motivated by the bounds obtained in Theorem 5.9 of [19], we present two theorems concerning the decay of the connectivity function for Bernoulli percolation in a bounded degree graph G in the supercritical phase.…”

“…We also study the two-point connectivity function in graphs satisfying the general criterion above. In particular we investigate the possibility, originally discussed in [19], about the existence of percolation processes in infinite graphs for which the finite connectivity decays nonexponentially even in the highly supercritical phase. Indeed, given a graph G without bi-infinite geodesics which falls in the class satisfying our new criterion, we show that the decay of the finite connectivity may be not exponential, presenting an example.…”

Percolation transition for random forests in $d\geq 3$

“…This behaviour is different from that of Bernoulli bond percolation and more generally that of the random cluster model with q > 0. For these models the percolation probability is governed by Peierls' contours and is 1 − O((1 − p) 2d ) by [63,Remark 5.10].…”