Exact distributions of cover times for $N$ independent random walkers in one dimension

Abstract

We study the probability density function (PDF) of the cover time $t_c$ of a finite interval of size $L$, by $N$ independent one-dimensional Brownian motions, each with diffusion constant $D$. The cover time $t_c$ is the minimum time needed such that each point of the entire interval is visited by at least one of the $N$ walkers. We derive exact results for the full PDF of $t_c$ for arbitrary $N\ge 1$, for both reflecting and periodic boundary conditions. The PDFs depend explicitly on $N$ and on the boundary conditions. In the limit of large $N$, we show that $t_c$ approaches its average value $⟨t_c⟩\approx L^2/(16D\ln N)$, with fluctuations vanishing as $1/(\ln N{)}^2$. We also compute the centered and scaled limiting distributions for large $N$ for both boundary conditions and show that they are given by nontrivial $N$-independent scaling functions.

5+2 pages, 4 figures

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