Statistical properties of sites visited by independent random walks
E. Ben-Naim and P.L. Krapivsky
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.
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