A short proof of free energy limit of two spin spherical Sherrington-Kirkpatrick model at any temperature

TL;DR

This paper provides a concise proof of the free energy limit for the spherical 2-spin Sherrington-Kirkpatrick model across all temperatures, utilizing Laplace approximation and eigenvalue distribution properties.

Contribution

It introduces a novel proof for the low-temperature case of the free energy limit, extending previous high-temperature results to all temperature regimes.

Findings

Validates the free energy limit at all temperatures.

Provides a general proof adaptable to various interaction matrices.

Extends the understanding of the spherical 2-spin SK model's thermodynamic behavior.

Abstract

In this paper, we present a short proof of the limit of free energy of spherical 2 spin Sherrington-Kirkpatrick (SSK) model without external field. This proof works for all temperatures and is based on the Laplace method of integration and considering a discrete approximation of the whole space. This proof is general enough and can be adapted to any interaction matrix where the eigenvalue distribution has some nice properties. The proof in the high-temperature case is the same as the proof given in [BK19]. However, the low-temperature case is new.