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 to it associated, 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.
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Franz et al. (2004) studied this question.