Irreducible free energy expansion and overlaps locking in mean field spin glasses

TL;DR

This paper develops a diagrammatic cavity field expansion near the critical temperature for mean field spin glasses, providing insights into overlap fluctuations, Ghirlanda-Guerra relationships, and the connection between free energy complexity and entropy.

Contribution

It introduces a novel diagrammatic formulation for cavity field expansion that simplifies the analysis of overlap fluctuations and the Ghirlanda-Guerra relationships in mean field spin glasses.

Findings

Derived a simple theory for overlap fluctuations.

Connected symmetry constraints to thermodynamic stability.

Expanded free energy highlighting the relation between solution complexity and entropy.

Abstract

We introduce a diagrammatic formulation for a cavity field expansion around the critical temperature. This approach allows us to obtain a theory for the overlap's fluctuations and, in particular, the linear part of the Ghirlanda-Guerra relationships (GG) (often called Aizenman-Contucci polynomials (AC)) in a very simple way. We show moreover how these constraints are "superimposed" by the symmetry of the model with respect to the restriction required by thermodynamic stability. Within this framework it is possible to expand the free energy in terms of these irreducible overlaps fluctuations and in a form that simply put in evidence how the complexity of the solution is related to the complexity of the entropy.