A Dynamic Programming Approach to the Parisi Functional

TL;DR

This paper introduces an elementary stochastic dynamic programming approach to analyze the Parisi functional in mean field spin glasses, providing new proofs of its properties without relying on complex cascade or transformation techniques.

Contribution

It offers a novel, simplified method to study the Parisi functional using PDE and stochastic programming, avoiding traditional complex tools.

Findings

Proves strict convexity of the Parisi functional.

Provides a new derivation of properties of the Parisi PDE.

Avoids use of Ruelle Cascades and Cole-Hopf transformations.

Abstract

G.Parisi predicted an important variational formula for the thermodynamic limit of the intensive free energy for a class of mean field spin glasses. In this paper, we present an elementary approach to the study of the Parisi functional using stochastic dynamic programing and semi-linear PDE. We give a derivation of important properties of the Parisi PDE avoiding the use of Ruelle Probability Cascades and Cole-Hopf transformations. As an application, we give a simple proof of the strict convexity of the Parisi functional, which was recently proved by Auffinger and Chen in [2].