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

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.