Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

TL;DR

This paper investigates the dynamics of disordered Ising spin systems on random graphs with finite connectivity, developing a generating functional approach to understand single-spin path probabilities and their evolution.

Contribution

It introduces a novel generating functional analysis for finite-connectivity disordered Ising models, extending understanding beyond fully connected systems.

Findings

Exact calculations for initial time steps of dynamics.

Macroscopic laws simplify for asymmetric graphs.

Simulation results validate theoretical predictions.

Abstract

We study the dynamics of bond-disordered Ising spin systems on random graphs with finite connectivity, using generating functional analysis. Rather than disorder-averaged correlation and response functions (as for fully connected systems), the dynamic order parameter is here a measure which represents the disorder averaged single-spin path probabilities, given external perturbation field paths. In the limit of completely asymmetric graphs our macroscopic laws close already in terms of the single-spin path probabilities at zero external field. For the general case of arbitrary graph symmetry we calculate the first few time steps of the dynamics exactly, and we work out (numerical and analytical) procedures for constructing approximate stationary solutions of our equations. Simulation results support our theoretical predictions.