Un-inverting the Parisi formula

TL;DR

This paper explores the inverted variational principle in mean-field spin glasses, providing a new stochastic-control representation of the Parisi functional through convex duality and martingale optimization.

Contribution

It introduces a novel stochastic-control formulation of the Parisi functional, transforming the limit free energy into a supremum over martingales, clarifying the inverted variational principle.

Findings

Rewrites the limit free energy as a supremum over martingales.

Uses convex duality to connect the Parisi functional with stochastic control.

Provides new insights into the structure of mean-field spin glasses.

Abstract

The free energy of any system can be written as the supremum of a functional involving an energy term and an entropy term. Surprisingly, the limit free energy of mean-field spin glasses is expressed as an infimum instead, a phenomenon sometimes called an inverted variational principle. Using a stochastic-control representation of the Parisi functional and convex duality arguments, we rewrite this limit free energy as a supremum over martingales in a Wiener space.