The full replica symmetry breaking solution in mean-field spin glass models

TL;DR

This thesis extends the Parisi full replica symmetry breaking solution to Ising spin glasses on random regular graphs using a new martingale approach, leading to a well-defined variational formulation and self-consistency equations.

Contribution

Introduces a martingale approach to extend the Parisi solution to random regular graphs, overcoming previous limitations and providing a new variational framework.

Findings

Development of a variational free energy functional for the model

Representation of the functional through backward stochastic differential equations

Derivation of self-consistency equations for order parameters

Abstract

This thesis focus on the extension of the Parisi full replica symmetry breaking solution to the Ising spin glass on a random regular graph. We propose a new martingale approach, that overcomes the limits of the Parisi-M\'ezard cavity method, providing a well-defined formulation of the full replica symmetry breaking problem in random regular graphs. We obtain a variational free energy functional, defined by the sum of two variational functionals (auxiliary variational functionals), that are an extension of the Parisi functional of the Sherrington-Kirkpatrick model. We study the properties of the two variational functionals in detailed, providing representation through the solution of a proper backward stochastic differential equation, that generalize the Parisi partial differential equation. Finally, we define the order parameters of the system and get a set of self-consistency equations for the order parameters and free energy.