The random-cluster model on the complete graph

Abstract

The random-cluster model of Fortuin and Kasteleyn contains as special cases the percolation, Ising, and Potts models of statistical physics. When the underlying graph is the complete graph onn vertices, then the associated processes are called ‘mean-field’. In this study of the mean-field random-cluster model with parametersp=λ/n andq, we show that its properties for any value ofq∈(0, ∞) may be derived from those of an Erdős-Renyi random graph. In this way we calculate the critical pointλ c (q) of the model, and show that the associated phase transition is continuous if and only ifq≦2. Exact formulae are given forλ C (q), the density of the largest component, the density of edges of the model, and the ‘free energy’. This work generalizes earlier results valid for the Potts model, whereq is an integer satisfyingq≧2. Equivalent results are obtained for a ‘fixed edge-number’ random-cluster model. As a consequence of the results of this paper, one obtains large-deviation theorems for the number of components in the classical random-graph models (whereq=1).