The probability of intersection of independent random walks in four dimensions

LetS ₁ andS ₂ be independent simple random walks of lengthn inZ ⁴ starting at 0 andx ₀ respectively. If |x ₀|₂n, 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.Research supported by National Science Foundation grant MCS-8002758