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 the associated processes are called 'mean-field'. In this study of the mean-field random-cluster model with parameters p = λ/n and q, we show that its properties for any value of q ∈ (0, ∞) may be derived from those of an Erdos-Rényi random graph. In this way we calculate the critical point λc(q) of the model, and show that the associated phase transition is continuous if and only if q ≦ 2. Exact formulae are given for λc(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, where q is an integer satisfying q ≧ 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).
Probability Theory and Related Fields
Bollobás, B., Grimmett, G., & Janson, S. (1996). The random-cluster model on the complete graph. Probability Theory and Related Fields, 104 (3), 283-317. https://doi.org/10.1007/bf01213683