The full replica symmetry breaking in the Ising spin glass on random regular graph

TL;DR

This paper develops a new mathematical framework for understanding the complex behavior of Ising spin glasses on random regular graphs, extending the full replica symmetry breaking scheme with a novel martingale approach.

Contribution

It introduces a martingale-based method to formulate the full replica symmetry breaking problem in random regular graphs, surpassing previous cavity methods.

Findings

Defined order parameters and derived self-consistency equations.

Provided a well-defined mathematical formulation for the problem.

Laid groundwork for future physical interpretation and quantitative analysis.

Abstract

In this paper, we extend the full replica symmetry breaking scheme 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. Finally, we define the order parameters of the system and get a set of self-consistency equations for the order parameters and the free energy. We face up the problem only from a technical point of view: the physical meaning of this approach and the quantitative evaluation of the solution of the self-consistency equations will be discussed in next works.