A duality principle in spin glasses

TL;DR

This paper establishes a duality principle linking free energies of Hamiltonians and their squared interactions in spin glasses, offering a new interpretation of the Parisi formula as an inverted variational principle.

Contribution

It introduces a duality principle for spin glasses connecting free energies of Hamiltonians and their squared interactions, with implications for the Parisi formula.

Findings

Duality principle valid for a large class of disordered systems.

Provides an interpretation of the Parisi formula as an inverted variational principle.

Establishes the relation under the assumption of concavity in the squared temperature parameter.

Abstract

We prove a duality principle that connects the thermodynamic limits of the free energies of the Hamiltonians and their squared interactions. Under the main assumption that the limiting free energy is concave in the squared temperature parameter, we show that this relation is valid in a large class of disordered systems. In particular, when applied to mean field spin glasses, this duality provides an interpretation of the Parisi formula as an inverted variational principle, establishing a prediction of Guerra.