Long-time Behavior for the Stochastic Ising Model with Unbounded Random Couplings

TL;DR

This paper investigates the long-time decay behavior of the one-dimensional stochastic Ising model with unbounded random couplings, revealing a transition from exponential to power law or stretched exponential decay.

Contribution

It demonstrates that unbounded couplings in the model lead to non-exponential decay behaviors, extending understanding beyond the case of bounded couplings.

Findings

Exponential decay occurs with bounded couplings.

Unbounded couplings cause power law or stretched exponential decay.

Decay behavior depends on the distribution of couplings.

Abstract

We consider the ferromagnetic Ising model with Glauber spin flip dynamics in one dimension. The external magnetic field vanishes and the couplings are i.i.d. random variables. If their distribution has compact support, the disorder averaged spin auto-correlation function has an exponential decay in time. We prove that, if the couplings are unbounded, the decay switches to either a power law or a stretched exponential, in general.