Convergence results and sharp estimates for the voter model interfaces

We study the evolution of the interface for the one-dimensional voter model. We show that
if the random walk kernel associated with the voter model has finite
th moment for some gamma > 3, then the evolution of the interface boundaries converge weakly to a Brownian motion
under diffusive scaling. This extends recent work of Newman, Ravishankar and Sun. Our result is optimal in the sense that finite th moment is necessary for this convergence for
all gamma in (0, 3). We also obtain relatively sharp estimates for the tail distribution of the size
of the equilibrium interface, extending earlier results of Cox and Durrett, and Belhaouari,
Mountford and Valle