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

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