Paper detail

One-dimensional Voter Model Interface Revisited

We consider the voter model on Z, starting with all 1&#39;s to the left of the origin and all 0&#39;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&#39;s and 0&#39;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&#39;s appearing deep within the regions of all 0&#39;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.