Spin Glass Computations and Ruelle's Probability Cascades

TL;DR

This paper investigates the Parisi functional within Ruelle's Probability Cascades, developing computation techniques to analyze its properties, connecting it to the Aizenman-Sims-Starr principle, and rederiving the Almeida-Thouless line.

Contribution

It introduces new computational methods for the RPC formulation of the Parisi functional and links it to key variational principles and phase transition lines in spin glass models.

Findings

Derived continuity and monotonicity properties of the Parisi functional.

Connected the RPC-based approach to the Aizenman-Sims-Starr variational principle.

Rederived the Almeida-Thouless line using RPC techniques.

Abstract

We study the Parisi functional, appearing in the Parisi formula for the pressure of the SK model, as a functional on Ruelle's Probability Cascades (RPC). Computation techniques for the RPC formulation of the functional are developed. They are used to derive continuity and monotonicity properties of the functional retrieving a theorem of Guerra. We also detail the connection between the Aizenman-Sims-Starr variational principle and the Parisi formula. As a final application of the techniques, we rederive the Almeida-Thouless line in the spirit of Toninelli but relying on the RPC structure.