Nature vs. Nurture: Dynamical Evolution in Disordered Ising Ferromagnets

TL;DR

This paper investigates the predictability and dynamical evolution of disordered ferromagnets under zero-temperature Glauber dynamics, revealing how disorder influences magnetization trajectories and twin overlap in various models.

Contribution

It provides a comprehensive theoretical and numerical analysis of the dynamical trajectories of disordered ferromagnets, highlighting the effects of disorder distributions and geometry on system evolution.

Findings

Mean-field with light-tailed couplings matches homogeneous models.

Heavy-tailed couplings retain disorder effects in the limit.

Twin overlap initially decorrelates then converges due to ferromagnetic drift.

Abstract

We study the predictability of zero-temperature Glauber dynamics in various models of disordered ferromagnets. This is analyzed using two independent dynamical realizations with the same random initialization (called twins). We derive, theoretically and numerically, trajectories for the evolution of the normalized magnetization and twin overlap as the system size tends to infinity. The systems we treat include mean-field ferromagnets with light-tailed and heavy-tailed coupling distributions, as well as highly-disordered models with a variety of other geometries. In the mean-field setting with light-tailed couplings, the disorder averages out and the limiting trajectories of the magnetization and twin overlap match those of the homogenous Curie--Weiss model. On the other hand, when the coupling distribution has heavy tails, or the geometry changes, the effect of the disorder persists in the thermodynamic limit. Nonetheless, qualitatively all such random ferromagnets share a similar time evolution for their twin overlap, wherein the two twins initially decorrelate, before either partially or fully converging back together due to the ferromagnetic drift.