One-dimensional Voter Model Interface Revisited

Abstract

We consider the voter model on $\mathbb{Z}$, 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 $1_{x< B_t} dx$, $t>0$.