Properties of the full replica symmetry breaking free energy functional of the Ising spin glass on random regular graph

TL;DR

This paper studies the full replica symmetry breaking free energy functional for the Ising spin glass on a random regular graph, showing it improves finite-RSB approximations and can be represented via a backward stochastic differential equation.

Contribution

It proves the full-RSB functional offers an improvement over finite-RSB methods and provides a stochastic differential equation representation for analysis.

Findings

Full-RSB functional improves finite-RSB approximations.

Representation as a unique solution to a backward stochastic differential equation.

Potential applications in calculus of variations and stochastic control.

Abstract

We analyze the full replica symmetry breaking (full--RSB) free energy functional for the Ising spin glass on a random regular graph proposed by the author in \cite{MyPaper}. We prove that the full--RSB formulation provides an improvement over any replica symmetry breaking approximation with a finite number of steps (finite--RSB), based on the M\'ezard-Parisi ansatz \cite{ParMezRRG1}. We provide a representation of that functional as the unique solution to a well-posed backward stochastic differential equation. This stochastic formulation enables a refined analysis of the functional and the computation of the derivatives with respect to the order parameters of the model. The techniques developed here hold potential interest for broader areas such as calculus of variations, stochastic optimal control, and functional analysis.