THE THERMODYNAMICS OF THE CURIE-WEISS MODEL WITH RANDOM COUPLINGS

We study the Curie-Weiss version of an Ising spin system with random, positively biased couplings. In particular, the case where the couplings epsilon(ij) take the values one with probability p and zero with probability 1 - p, which describes the Ising model on a random graph, is considered. We prove that if p is allowed to decrease with the system size N in such a way that Np(N) up infinity as N up infinity, then the free energy converges (after trivial rescaling) to that of the standard Curie-Weiss model, almost surely. Similarly, the induced measures on the mean magnetizations converge to those of the Curie-Weiss model. Generalizations of this result to a wide class of distributions are detailed.