The Kac limit for finite-range spin glasses

TL;DR

This paper proves that in the Kac limit, the free energy of finite-range spin glasses converges to the Sherrington-Kirkpatrick model, suggesting a pathway for mean-field approximations in spin glass systems.

Contribution

It establishes the convergence of finite-range spin glass free energy to the SK model in the Kac limit under mild assumptions.

Findings

Convergence of free energy to SK model in the Kac limit

Supports expansion around mean-field theory for spin glasses

Provides a foundation for analyzing finite-range spin glasses

Abstract

We consider a finite range spin glass model in arbitrary dimension, where the strength of the two-body coupling decays to zero over some distance $\gamma^{-1}$. We show that, under mild assumptions on the interaction potential, the infinite-volume free energy of the system converges to that of the Sherrington-Kirkpatrick one, in the Kac limit $\gamma\to0$. This could be a first step toward an expansion around mean field theory, for spin glass systems.