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

TL;DR

This paper derives exact probability density functions for the cover time of a finite interval by multiple independent one-dimensional Brownian motions, revealing how these distributions depend on the number of walkers and boundary conditions.

Contribution

It provides the first exact formulas for the full PDF of cover times for any number of walkers and boundary conditions, including large N asymptotics.

Findings

Exact PDFs derived for all N and boundary conditions.

As N increases, the mean cover time scales as L^2/(16 D ln N).

Fluctuations diminish as 1/(ln N)^2 for large N.

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 $N \geq 1$, 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 $\langle t_c \rangle \approx L^2/(16\, D \, \ln N)$, 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.