THE RANDOM-CLUSTER MODEL ON THE COMPLETE GRAPH

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 on n vertices, then t he associated processes are called 'mean-field'. In this Study Of the mean-field random-cluster model with parameters p = lambda/n and q, we show that its properties for any value of q is an element of (0, infi nity) may be derived from those of an Erdos-Renyi random graph. In thi s way we calculate the critical point lambda(c)(q) of the model, and s how that the associated phase transition is continuous if and only if q less than or equal to 2. Exact formulae are given for lambda(c)(q), the density of the largest component, the density of edges of the mode l, and the 'free energy'. This work generalizes earlier results valid for the Potts model, where q is an integer satisfying q greater than o r equal to 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 (where q = 1).