title: 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≥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 〈t_{c}〉≈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.
authors.affiliation: LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.
authors.forename: Satya N
authors.initials: SN
authors.lastname: Majumdar
authors.affiliation: Raman Research Institute, Bangalore 560080, India.
authors.forename: Sanjib
authors.initials: S
authors.lastname: Sabhapandit
authors.affiliation: LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.
authors.forename: Grégory
authors.initials: G
authors.lastname: Schehr