Notes on the p-spin glass studied via Hamilton-Jacobi and Smooth-Cavity techniques

TL;DR

This paper investigates the p-spin glass model using Hamilton-Jacobi and smooth cavity techniques, deriving free energy expressions and overlap identities, advancing understanding of disordered systems' thermodynamics.

Contribution

It introduces a comprehensive analysis of the p-spin glass model with Ising spins using Hamilton-Jacobi and smooth cavity methods, providing new free energy formulas and overlap relations.

Findings

Derived RS and 1RSB free energy expressions

Established self-consistent overlap relations

Connected overlap identities with internal energy and entropy

Abstract

In these notes, we continue our investigation of classical toy models of disordered statistical mechanics through various techniques recently developed and tested mainly on the paradigmatic SK spin glass. Here we consider the p-spin-glass model with Ising spins and interactions drawn from a normal distribution N[0,1]. After a general presentation of its properties (e.g. self-averaging of the free energy, existence of a suitable thermodynamic limit), we study its equilibrium behavior within the Hamilton-Jacobi framework and the smooth cavity approach. Through the former we find both the RS and the 1RSB expressions for the free energy, coupled with their self-consistent relations for the overlaps. Through the latter, we recover these results as irreducible expression, and we study the generalization of the overlap polynomial identities suitable for this model; a discussion on their deep connection with the structure of the internal energy and the entropy closes the investigation.