Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with ferromagnetic interaction

TL;DR

This paper analyzes the fluctuations of free energy in a spherical spin system combining SK and Curie-Weiss interactions, revealing phase transitions and limiting distributions across parameters.

Contribution

It introduces the first analysis of free energy fluctuations in a spherical SK model with ferromagnetic interaction, including a CLT for linear statistics of spiked matrices.

Findings

Limiting distributions of free energy away from critical points

Identification of a phase transition related to the largest eigenvalue

Establishment of a CLT for linear statistics of spiked matrices

Abstract

We consider a spherical spin system with pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian with ferromagnetic Curie-Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the coupling constant. We compute the limiting distributions of the free energy for all parameters away from the critical values. The zero temperature case corresponds to the well-known phase transition of the largest eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step, we establish a central limit theorem for the linear statistics of rank 1 spiked random symmetric matrices.