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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)\\sim t^{d/2} for large t (the N dependence is only in the prefactor). For 2<d<d_c(N), W_N(t)\\sim t^{\\nu} where the exponent \\nu= N-d(N-1)/2 varies with N and d. 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Tamm, Satya N. Majumdar","submitted_at":"2012-06-27T06:43:26Z","abstract_excerpt":"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)\\sim t^{d/2} for large t (the N dependence is only in the prefactor). For 2<d<d_c(N), W_N(t)\\sim t^{\\nu} where the exponent \\nu= N-d(N-1)/2 varies with N and d. 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