Approximation schemes for the dynamics of diluted spin models: the Ising ferromagnet on a Bethe lattice

TL;DR

This paper introduces approximation schemes for analyzing the dynamics of diluted spin models, specifically applied to the Ising ferromagnet on a Bethe lattice, providing accurate finite-time predictions and insights into critical behavior.

Contribution

It develops a hierarchy-based approximation method for spin dynamics and demonstrates its effectiveness on the Bethe lattice Ising model, connecting it to dynamical replica theory.

Findings

Exact long-time behavior predictions

Improved finite-time dynamic predictions

Insights into critical region and correlation functions

Abstract

We discuss analytical approximation schemes for the dynamics of diluted spin models. The original dynamics of the complete set of degrees of freedom is replaced by a hierarchy of equations including an increasing number of global observables, which can be closed approximately at different levels of the hierarchy. We illustrate this method on the simple example of the Ising ferromagnet on a Bethe lattice, investigating the first three possible closures, which are all exact in the long time limit, and which yield more and more accurate predictions for the finite-time behavior. We also investigate the critical region around the phase transition, and the behavior of two-time correlation functions. We finally underline the close relationship between this approach and the dynamical replica theory under the assumption of replica symmetry.