Probability of hitting a far point for the voter model

We consider a voter model on the integer lattice started with a single one at the origin. In dimensions 2 and 3, we establish the precise asymptotic behaviour of the probability for the voter model to hit a distant point. We use the scaling limit of the voter model started from a single one in terms of super-Brownian motion under its excursion measure. This invariant principle was proved by Bramson Cox and Le Gall, as a consequence of a theorem of Cox, Durrett and Perkins. We also derive less precise estimates in dimension less than 4.