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.

arXiv comment5+2 pages, 4 figures

References in corpus (16)

• Intermittent search strategies
• Persistence and First-Passage Properties in Non-equilibrium Systems
• First passages for a search by a swarm of independent random searchers
• Stochastic Search with Poisson and Deterministic Resetting
• Random Convex Hulls and Extreme Value Statistics
• Mortality, Redundancy, and Diversity in Stochastic Search
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• Record Statistics for Multiple Random Walks
• Exact distributions of the number of distinct and common sites visited by N independent random walkers
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• Convex hull of a Brownian motion in confinement
• Number of Common Sites Visited by N Random Walkers
• Random walks on networks: cumulative distribution of cover time
• Maximal distance travelled by N vicious walkers till their survival
• Records for the number of distinct sites visited by a random walk on the fully-connected lattice
• Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

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