  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.
