Convex Replica Simmetry Breaking From Positivity and Thermodynamic Limit

TL;DR

This paper demonstrates that in a Gaussian energy model with positivity constraints, the thermodynamic limit enforces a convex replica symmetry breaking structure in the covariance matrix, aligning with Parisi's scheme.

Contribution

It establishes a link between positivity conditions, the thermodynamic limit, and the emergence of convex replica symmetry breaking in Gaussian models.

Findings

Covariance matrix must follow Parisi RSB scheme under positivity constraints

Thermodynamic limit enforces convexity in the covariance matrix

Connects positivity conditions with replica symmetry breaking structure

Abstract

Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity condition on the matrix elements.