Exact distributions of the number of distinct and common sites visited by N independent random walkers

TL;DR

This paper derives exact probability distributions for the number of distinct and common sites visited by N independent one-dimensional random walkers, revealing their asymptotic behaviors and verifying results with simulations.

Contribution

It provides the first exact distributions for these quantities and maps them onto extreme value problems, advancing understanding of multi-walker random walk statistics.

Findings

Exact distributions for S_N(t) and W_N(t) are derived.

Asymptotic behaviors for large N are characterized.

Results are confirmed through numerical simulations.

Abstract

We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\sqrt{t} \propto 2 \sqrt{\log N} + \widetilde{s}/(2 \sqrt{\log N}) and W_N(t)/\sqrt{t} \propto \widetilde{w}/N where \widetilde{s} and \widetilde{w} are random variables whose probability density functions (pdfs) are computed exactly and are found to be non trivial. We verify our results through direct numerical simulations.