The probability of intersection of independent random walks in four dimensions

Abstract

LetS 1 andS 2 be independent simple random walks of lengthn inZ 4 starting at 0 andx 0 respectively. If |x 0|2≈n, it is shown that the probability that the paths intersect is of order (logn)−1. Ifx 0=0, it is shown that the probability of no intersection of the paths decays no faster than (logn)−1 and no slower than (logn)−1/2. It is conjectured that (logn)−1/2 is the actual decay rate.