The Kac limit for diluted spin glasses

We study diluted spin glass models in arbitrary dimension, where each spin interacts with a finite number of other spins chosen at random with a probability decaying to zero over some distance ¿-1. For systems with pairwise interactions we show that the infinite-volume free energy converges to that of the mean-field Viana–Bray model,1 in the Kac limit ¿¿0. For p-spin like models we get only one bound: the free-energy is bounded from above by the one of the mean-field diluted p-spin.