A Note on the Free Energy of the Coupled System in the Sherrington - Kirkpatrick Model

#### D. Panchenko

2005, v.11, №1, 19-36

ABSTRACT

In this paper we consider a system of spins that consists of two configurations $\vec{\sigma}^1,\vec{\sigma}^2\in\Sigma_N=\{-1,+1\}^N$ with Gaussian Hamiltonians $H_N^1(\vec{\sigma}^1)$ and $H_N^2(\vec{\sigma}^2)$ 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.

Keywords: spin glasses,Sherrington - Kirkpatrick model