Distinct and Common Sites Visited by N Random Walkers
Laboratoire de Physique Theorique et Modeles Statistique, France
Number of distinct sites visited by a single random walker in time t in d-dimensions is a classic problem in random walk theory with several applications. When there are multiple (N) walkers, both the distinct and the common sites visited by all the walkers are of interest. In this talk, I will describe recent exact results on the mean number of common sites visited by N independent random walkers (each of length t) on a d-dimensional lattice. We find an interesting and somewhat unusual phase transition in the N-d plane. There are three distinct phases characterizing different asymptotic growth of the number of common sites with t. These three phases in the N-d plane are separated by two critical lines. In one dimension, the full distribution of both the distinct and the common sites for N walkers can be computed exactly. Implications and applications of these results will be discussed.