STRICT INEQUALITY FOR CRITICAL-VALUES OF POTTS MODELS AND RANDOM-CLUSTER PROCESSES

We prove that the critical value beta(c) of a ferromagnetic Potts model is a strictly decreasing function of the strengths of interaction of the process. This is achieved in the (more) general context of the random-cluster representation of Fortuin and Kasteleyn, by deriving and utilizing a formula which generalizes the technique known in percolation theory as Russo's formula. As a byproduct of the method, we present a general argument for showing that, at any given point on the critical surface of a multiparameter process, the values of a certain critical exponent do not depend on the direction of approach of that point. Our results apply to all random-cluster processes satisfying the FKG inequality.