Number of Common Sites Visited by N Random Walkers

Satya N. Majumdar; Mikhail V. Tamm

doi:10.1103/physreve.86.021135

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arXiv · 1206.6184

Number of Common Sites Visited by N Random Walkers

Satya N. Majumdar·Mikhail V. Tamm

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Abstract

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_c(N), W_N(t) approaches a constant as t\to \infty. Exactly at the critical dimensions there are logaritmic corrections: for d=2, we get W_N(t)\sim t/[\ln t]^N, while for d=d_c(N), W_N(t)\sim \ln t for large t. Our analytical predictions are verified in numerical simulations.

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BibTeX RIS

Satya N. Majumdar, Mikhail V. Tamm. 2012-06-27. Number of Common Sites Visited by N Random Walkers. https://doi.org/10.1103/physreve.86.021135

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