Dynamical replica analysis of disordered Ising spin systems on finitely connected random graphs

TL;DR

This paper extends dynamical replica and cavity methods to analyze the macroscopic dynamics of disordered Ising spin systems on finite connectivity random graphs, with applications to spin glasses and error-correcting codes.

Contribution

It generalizes existing theoretical techniques to systems with strong disorder and heterogeneity, enabling analysis of their macroscopic dynamics.

Findings

Theoretical predictions match Monte Carlo simulations.

Effective description of magnetization and energy dynamics.

Application to spin glasses and Sourlas codes demonstrates versatility.

Abstract

We study the dynamics of macroscopic observables such as the magnetization and the energy per degree of freedom in Ising spin models on random graphs of finite connectivity, with random bonds and/or heterogeneous degree distributions. To do so we generalize existing implementations of dynamical replica theory and cavity field techniques to systems with strongly disordered and locally tree-like interactions. We illustrate our results via application to the dynamics of e.g. $\pm J$ spin-glasses on random graphs and of the overlap in finite connectivity Sourlas codes. All results are tested against Monte Carlo simulations.