Number of common sites visited by $N$ random walkers

We compute analytically the mean number of common sites, WN(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 WN(t). These three regimes are separated by two critical lines d=2 and d=dc(N)=2N/(N−1) in the (N-d) plane. For d<2, WN(t)∼td/2 for large t (the N dependence is only in the prefactor). For 2<d<dc(N), WN(t)∼tν where the exponent ν=N−d(N−1)/2 varies with N and d. For d>dc(N), WN(t)→const as t→∞. Exactly at the critical dimensions there are logarithmic corrections: for d=2, we get WN(t)∼t/[lnt]N, while for d=dc(N), WN(t)∼lnt for large t. Our analytical predictions are verified in numerical simulations.