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 couplingsɛij take the values one with probabilityp and zero with probability 1 −p, which describes the Ising model on a random graph, is considered. We prove that ifp is allowed to decrease with the system sizeN in such a way thatNp(N) ↑ ∞ asN ↑ ∞, 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.