1. Center for Information Technology
2. National Institute of Child Health and human Development
National Institutes of Health
Bethesda, Md. 20892
April 12, 2000
3. Permanent address: Karpov Institute of Physical Chemistry, 10 Vorontsovo Pole Street, 103064 Moscow K-64, Russia.
We consider an infinite number of non-interacting lattice random walkers with the goal of determining statistical properties of the time, out of a total time T, that a single site has been occupied by n random walkers. Initially the random walkers are assumed uniformly distributed on the lattice except for the target site at the origin, which is unoccupied. The random walk model is taken to be continuous-time random walk and the pausing-time density at the target site is allowed to differ from the pausing-time density at other sites. We calculate the dependence of the mean time of occupancy by n random walkers as a function of n> and the observation time T. We also find the variance for the cumulative time during which the site is unoccupied. The large-T behavior of the variance differs according as the random walk is transient or recurrent. It is shown that the variance is proportional to T at large T in three or more dimensions, it is proportional to T3/2 in one dimension and to TlnT in two dimensions.