Finite-range spin glasses in the Kac limit: free energy and local observables

We study a finite-range spin-glass model in arbitrary dimension, where the intensity of the coupling between spins decays to zero over some distance γ⁻¹. We prove that, under a positivity condition for the interaction potential, the infinite-volume free energy of the system converges to that of the Sherrington–Kirkpatrick model, in the Kac limit γ → 0. We study the implication of this convergence for the local order parameter, i.e., the local overlap distribution function and a family of susceptibilities associated with it, and we show that locally the system behaves like its mean field analogue. Similar results are obtained for models with p-spin interactions. Finally, we discuss a possible approach to the problem of the existence of long-range order for finite γ, based on a large deviation functional for overlap profiles. This will be developed in future work.