Statistical properties of sites visited by independent random walks

The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0,t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the ordering probabilities in spatial dimensions d=1,2,3,4. We also study the evolution of the number of ties between the number of visited sites. We show analytically that the average number of ties increases as $a_1\ln t$ with $a_1=0.970508$ in one dimension and as $(\ln t)^2$ in two dimensions.