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 with $N$ independent random walkers. Using this mapping, we compute exactly their probability distributions $P_N^d(S,t)$ and $P_N^c(W,t)$ 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 $S_N(t)/\sqrt{t}\propto 2\sqrt{\log \hspace{0.17em}N}+\tilde{s}/(2\sqrt{\log \hspace{0.17em}N})$ and $W_N(t)/\sqrt{t}\propto \tilde{w}/N$ where $\tilde{s}$ and $\tilde{w}$ are random variables whose probability density functions are computed exactly and are found to be nontrivial. We verify our results through direct numerical simulations.

• Received 11 February 2013

DOI:https://doi.org/10.1103/PhysRevLett.110.220602