Rigorous approaches for spin glass and Gaussian spin glass with P-wise interactions

TL;DR

This paper rigorously analyzes P-spin and Gaussian P-spin glass models with polynomial interactions, confirming known results through transport and Guerra's methods, and establishing the intrinsic replica symmetry for P=2.

Contribution

It provides a rigorous derivation of the quenched pressure and self-consistency equations for P-spin glasses using transport and Guerra's techniques, validating previous non-rigorous results.

Findings

Recovered the same quenched pressure expressions as replica methods.

Confirmed the intrinsic replica symmetry for P=2.

Validated the models using two rigorous mathematical approaches.

Abstract

Purpose of this paper is to face up to P-spin glass and Gaussian P-spin model, i.e. spin glasses with polynomial interactions of degree P > 2. We consider the replica symmetry and first step of replica simmetry breaking assumptions and we solve the models via transport equation and Guerra's interpolating technique, showing that we reach the same results. \\ Thus, using rigorous approaches, we recover the same expression for quenched statistical pressure and self-consistency equation in both assumption found with other techniques, including the well-known \textit{replica trick} technique. \\ At the end, we show that for $P=2$ the Gaussian P-spin glass model is intrinsecally RS.