Occupancy of a single site by many random walkers

We consider an infinite number of noninteracting lattice random walkers wit h the goal of determining statistical properties of the time, out of a tota l time T, that a single site has been occupied by it random walkers. Initia lly the random walkers are assumed uniformly distributed on the lattice exc ept for the target site at the origin, which is unoccupied. The random-walk model is taken to be a continuous-time random walk and the pausing-time de nsity 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 unoccupi ed. The large-T behavior of the variance differs according as the random wa lk 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 T-3/2 i n one dimension and to T ln T in two dimensions.