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 $\vsi^1,\vsi^2\inΣ_N=\{-1,+1\}^N$ 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.

16 pages

References in corpus (3)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• An Extended Variational Principle for the SK Spin-Glass Model
• The infinite volume limit in generalized mean field disordered models

Cited by in corpus (1)

• Spin glass models from the point of view of spin distributions