Number of common sites visited by N random walkers

We compute analytically the mean number of common sites, W(N)(t), visited by N independent random walkers each of length t and all starting at the origin at t = 0 in d dimensions. We show that in the (N-d) plane, there are three distinct regimes for the asymptotic large-t growth of W(N)(t). These three regimes are separated by two critical lines d = 2 and d = d(c)(N) = 2N/(N-1) in the (N-d) plane. For d<2, W(N) (t) ~ t(d/2) for large t (the N dependence is only in the prefactor). For 2 < d < d(c)(N), W(N)(t) ~ t(ν) where the exponent ν = N-d(N-1)/2 varies with N and d. For d > d(c)(N), W(N)(t) → const as t → ∞. Exactly at the critical dimensions there are logarithmic corrections: for d=2, we get W(N)(t) ~ t/[ln t](N), while for d = d(c)(N), W(N)(t) ~ ln t for large t. Our analytical predictions are verified in numerical simulations.