Persistence in the Voter model: continuum reaction-diffusion approach

TL;DR

This paper analyzes the persistence probability in the voter model across different dimensions by mapping it to a reaction-diffusion system and deriving explicit asymptotic decay laws, validated through simulations.

Contribution

It introduces a continuum reaction-diffusion approach to compute voter model persistence probabilities and provides explicit decay laws for various dimensions.

Findings

Persistence decays as exp[-f_2(q)(ln t)^2] in 2D

Persistence decays as exp[-f_d(q)t^{(d-2)/2}] for 2<d<4

Persistence decays as exp[-f_4(q)t/ln t] in 4D and higher

Abstract

We investigate the persistence probability in the Voter model for dimensions d\geq 2. This is achieved by mapping the Voter model onto a continuum reaction-diffusion system. Using path integral methods, we compute the persistence probability r(q,t), where q is the number of ``opinions'' in the original Voter model. We find r(q,t)\sim exp[-f_2(q)(ln t)^2] in d=2; r(q,t)\sim exp[-f_d(q)t^{(d-2)/2}] for 2<d<4; r(q,t)\sim exp[-f_4(q)t/ln t] in d=4; and r(q,t)\sim exp[-f_d(q)t] for d>4. The results of our analysis are checked by Monte Carlo simulations.