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 of $\langle t_c\rangle \approx L^2/(16DlnN)$ with fluctuations vanishing as $1/{(lnN)}^2$. 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.

• Received 21 September 2016

DOI:https://doi.org/10.1103/PhysRevE.94.062131