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

TL;DR

This paper proves that finite-range spin glasses converge to the Sherrington-Kirkpatrick model in the Kac limit, showing local behavior similar to mean field theory and discussing implications for long-range order.

Contribution

It establishes the convergence of free energy and local observables of finite-range spin glasses to mean field models in the Kac limit, extending results to p-spin interactions.

Findings

Convergence of free energy to SK model in the Kac limit

Local overlap distribution matches mean field predictions

Discussion on large deviation approach for long-range order

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 $\gamma\to0$. 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.