# An invariance principle for biased voter model interfaces

## TL;DR

This paper studies the behavior of biased voter model interfaces in one dimension, showing that under weak bias and diffusive scaling, the interface converges to a drifted Brownian motion, extending known unbiased results.

## Contribution

It introduces a diffusive limit for biased voter model interfaces, incorporating drift, under finite second moment conditions, extending previous unbiased model analyses.

## Key findings

- Interface converges to a drifted Brownian motion in the diffusive limit.
- Most sites on either side of the interface are of the dominant type.
- Results depend on recent interface tightness theorems for biased models.

## Abstract

We consider one-dimensional biased voter models, where 1's replace 0's at a faster rate than the other way round, started in a Heaviside initial state describing the interface between two infinite populations of 0's and 1's. In the limit of weak bias, for a diffusively rescaled process, we consider a measure-valued process describing the local fraction of type 1 sites as a function of time. Under a finite second moment condition on the rates, we show that in the diffusive scaling limit there is a drifted Brownian path with the property that all but a vanishingly small fraction of the sites on the left (resp. right) of this path are of type 0 (resp. 1). This extends known results for unbiased voter models. Our proofs depend crucially on recent results about interface tightness for biased voter models.