Updating schemes in zero-temperature single-spin flip dynamics

TL;DR

This paper explores how different updating schemes affect the relaxation process in zero-temperature single-spin flip dynamics on a 1D lattice, revealing how relaxation times scale with system size and the nature of the dynamics.

Contribution

It introduces a systematic study of $c$-parallel updating schemes, analyzing their impact on relaxation times and establishing an empirical relation between inflow and outflow dynamics.

Findings

Relaxation times depend on the updating scheme parameter $c$.

An empirical formula relates relaxation times of inflow and outflow dynamics.

Original zero-temperature Glauber dynamics is identified as a critical case.

Abstract

In this paper we examine the role of the so called $c$-parallel updating schemes in relaxation from disordered states to the final ferromagnetic steady state. We investigate two zero-temperature single-spin flip dynamics on a one dimensional lattice of length $L$: inflow (i.e. generalized zero-temperature Glauber dynamics) and outflow opinion dynamics. The varying $c$ allows us to change the updating scheme from random sequential updating ($c=1/L$) to deterministic synchronous updating (for $c=1$). We show how the mean relaxation times depend on $c$ and scale with the system size $L$. Moreover, we empirically find an analytical formula for the ratio between mean relaxation times for inflow and outflow dynamics. Results obtained in this paper suggest that in some sense the original zero-temperature Glauber dynamics is a critical one among a broader class of inflow dynamics.