One-dimensional Voter Model Interface Revisited

TL;DR

This paper revisits the one-dimensional voter model, demonstrating that under finite second moment conditions, the rescaled interface converges to a Brownian motion, extending understanding of the model's scaling limits.

Contribution

It establishes the weak convergence of the measure-valued process for the voter model with finite second moment, broadening previous results that required higher moments.

Findings

Convergence of the interface to Brownian motion under finite second moment.

Tightness of the measure-valued process is proven for finite second moments.

Failure of convergence when moments are infinite below the third order.

Abstract

We consider the voter model on Z, starting with all 1's to the left of the origin and all 0's to the right of the origin. It is known that if the associated random walk kernel p has zero mean and a finite r-th moment for any r>3, then the evolution of the boundaries of the interface region between 1's and 0's converge in distribution to a standard Brownian motion (B_t)_{t>0} under diffusive scaling of space and time. This convergence fails when p has an infinite r-th moment for any r<3, due to the loss of tightness caused by a few isolated 1's appearing deep within the regions of all 0's (and vice versa) at exceptional times. In this note, we show that as long as p has a finite second moment, the measure-valued process induced by the rescaled voter model configuration is tight, and converges weakly to the measure-valued process 1_{x<B_t}dx, t>0.