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

Abstract

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 ${\gamma }^{-1}$. 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 $\gamma \to 0$. 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 $\gamma$, based on a large deviation functional for overlap profiles. This will be developed in future work.

19 pages, revtex4

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• Ordering and Broken Symmetry in Short-Ranged Spin Glasses
• The Kac limit for finite-range spin glasses
• Thermodynamical Limit for Correlated Gaussian Random Energy Models
• Quasi-equilibrium during aging of the 2D Edwards-Anderson model

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