Stretched Exponential Relaxation in the Biased Random Voter Model

TL;DR

This paper investigates how introducing random bias affects the relaxation dynamics of the voter model, showing that disorder accelerates relaxation beyond classical stretched exponential bounds, with proven optimality under certain conditions.

Contribution

It provides a rigorous analysis of the relaxation speed in biased voter models, establishing bounds and optimality results that extend understanding of disorder effects.

Findings

Disorder-averaged relaxation is faster than a stretched exponential with specific exponent.

Under certain conditions, the upper bound on relaxation speed is proven to be optimal.

The analysis leverages a key result by Donsker and Varadhan (1979).

Abstract

We study the relaxation properties of the voter model with i.i.d. random bias. We prove under mild condions that the disorder-averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent $d/(d+\alpha)$, where $0<\alpha\le 2$ depends on the transition rates of the non-biased voter model. Under an additional assumption, we show that the above upper bound is optimal. The main ingredient of our proof is a result of Donsker and Varadhan (1979).