Equilibrium interfaces of biased voter models

TL;DR

This paper extends the understanding of interface tightness in one-dimensional voter models to biased versions, showing convergence of equilibrium interface distributions as bias diminishes, with implications for biological population models.

Contribution

It proves interface tightness for biased voter models and demonstrates the convergence of their equilibrium interface distributions to the unbiased case as bias approaches zero.

Findings

Interface tightness holds for biased voter models.

Equilibrium interface distributions converge to the unbiased voter model as bias tends to zero.

An identity for the expected number of boundaries in the equilibrium interface is established.

Abstract

A one-dimensional interacting particle system is said to exhibit interface tightness if starting in an initial condition describing the interface between two constant configurations of different types, the process modulo translations is positive recurrent. In a biological setting, this describes two populations that do not mix, and it is believed to be a common phenomenon in one-dimensional particle systems. Interface tightness has been proved for voter models satisfying a finite second moment condition on the rates. We extend this to biased voter models. Furthermore, we show that the distribution of the equilibrium interface for the biased voter model converges to that of the voter model when the bias parameter tends to zero. A key ingredient is an identity for the expected number of boundaries in the equilibrium voter model interface, which is of independent interest.