Probability of hitting a distant point for the voter model started with a single one

2006/09/28 by Mathieu Merle, Merle, Mathieu
Mathematics · #60F17 #60G57 #60J80 #60K30 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F17 #msc:60G57 #msc:60J80 #msc:60K30

paper · pdf · doi:10.48550/arxiv.math/0609826

arxiv created 2006/09/28 · arxiv updated 2009/12/01

Abstract

The goal of this work is to find the asymptotics of the hitting probability of a distant point for the voter model on the integer lattice started from a single 1 at the origin. In dimensions 2 or 3, we obtain the precise asymptotic behavior of this probability. We use the scaling limit of the voter model started from a single 1 at the origin in terms of super-Brownian motion under its excursion measure. This invariance principle was stated by Bramson, Cox and Le Gall, as a consequence of a theorem of Cox, Durrett and Perkins. Less precise estimates are derived in dimensions greater than 4.

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