Time-delay Induced Dimensional Crossover in Voter Model

TL;DR

This paper studies how time delays in the voter model cause a crossover in the system's effective dimensionality, revealing a transition from higher to lower-dimensional behavior over time through analytical and numerical methods.

Contribution

It introduces a novel mapping of the delayed voter model dynamics to a first passage problem, uncovering the dimensional crossover phenomenon.

Findings

Time delay causes a transition from (d+1)-dimensional to d-dimensional scaling.

Analytical expressions for the crossover time scale are derived.

Numerical simulations confirm the theoretical predictions.

Abstract

We investigate the ordering dynamics of the voter model with time-delayed interactions. The dynamical process in the $d$-dimensional lattice is shown to be equivalent to the first passage problem of a random walker in the $(d+1)$-dimensional strip of a finite width determined by the delay time. The equivalence reveals that the time delay leads to the dimensional crossover from the $(d+1)$-dimensional scaling behavior at short time to the $d$-dimensional scaling behavior in the long time. The scaling property in both regimes and the crossover time scale are obtained analytically, which are confirmed with the numerical simulation results.