Abstract
We study the number of distinct sites and common sites visited by independent one dimensional random walkers, all starting at the origin, after time steps. We show that these two random variables can be mapped onto extreme value quantities associated with independent random walkers. Using this mapping, we compute exactly their probability distributions and for any value of in the limit of large time , where the random walkers can be described by Brownian motions. In the large limit one finds that and where and 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
© 2013 American Physical Society