A note on the free energy of the coupled system in the Sherrington-Kirkpatrick model

TL;DR

This paper investigates the free energy of a coupled spin system in the Sherrington-Kirkpatrick model, establishing the thermodynamic limit and providing a variational principle involving random overlap structures.

Contribution

It introduces a framework for analyzing the free energy of coupled SK model systems with fixed overlaps, extending existing variational principles.

Findings

Proves the existence of the thermodynamic limit of free energy for coupled systems.

Derives an analogue of the Aizenman-Sims-Starr variational principle for the coupled model.

Provides a characterization of the free energy via random overlap structures.

Abstract

In this paper we consider a system of spins that consists of two configurations $\vsi^1,\vsi^2\in\Sigma_N=\{-1,+1\}^N$ with Gaussian Hamiltonians $H_N^1(\vsi^1)$ and $H_N^2(\vsi^2)$ correspondingly, and these configurations are coupled on the set where their overlap is fixed $\{R_{1,2}=N^{-1}\sum_{i=1}^N \sigma_i^1\sigma_i^2 = u_N\}.$ We prove the existence of the thermodynamic limit of the free energy of this system given that $\lim_{N\to\infty}u_N = u\in[-1,1]$ and give the analogue of the Aizenman-Sims-Starr variational principle that describes this limit via random overlap structures.