# Fluctuation of the free energy of Sherrington-Kirkpatrick model with Curie-Weiss interaction: the paramagnetic regime

## TL;DR

This paper analyzes the free energy fluctuations of a mixed Sherrington-Kirkpatrick and Curie-Weiss spin model in the paramagnetic regime, showing asymptotic Gaussianity and spectral approximation methods for i.i.d. Rademacher spins.

## Contribution

It extends previous spherical models to i.i.d. Rademacher spins, proving Gaussian fluctuations and spectral approximation of free energy in the paramagnetic phase.

## Key findings

- Free energy is asymptotically Gaussian.
- Approximation by linear spectral statistics.
- Method relies on dense sub-graph conditioning.

## Abstract

We consider a spin system with pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by \citet{Baiklee16} and \citet{Baikleewu18} which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to \citet{Baiklee16} in the "paramagnetic regime" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by \citet{Ban16}. The proof of the approximation by the linear spectral statistics part is taken from \citet{Banerjee2017}.