Differential Inequalities for Potts and Random-Cluster Processes

Let θ(J) be the order parameter of a (ferromagnetic) Potts or randomcluster process with bond-variables J = (Je : e ∈ K). We discuss differential inequalities of the form ∂θ ∂Je ≤ α(J) ∂θ ∂Jf for all e, f ∈ K. Such inequalities may be established for all random-cluster processes that satisfy the FKG inequality, possibly in the presence of many-body interactions (subject to certain necessary and sufficient conditions on the sets of interactions). There are (at least) two principal consequences of this. First, for a process having ‘inverse-temperature’ β, the critical value βc = βc(J) is a strictly monotone function of J. Secondly, at any fixed point J lying on the critical surface of the process, the critical exponent of θ in the limit as J ↓ J is independent of the direction of approach of the limit. Such a conclusion should be valid for other critical exponents also; this amounts to a small amount of rigorous universality.