THE LATTICE COVERING TIME PROBLEM FOR SITES VISITED KAPPA-TIMES

The problem of the covering time for sites visited k-times is defined as the mean time taken by a random walk to visit each site of a lattice at least k times. We performed the investigation using Monte Carlo simulations over one dimensional lattices, of N sites, with periodic boundary conditions. Two different regions are investigated: N >> k >> 1 and k >> N >> 1. In the former region, we obtain a behaviour of the type t(k)/t1 = a infinity - Bk-0.35 + A(k) N-0.75, (a infinity < 1.6). In the latter region we obtain two possible behaviours: t(k) approximately k0.95 and t(k) approximately k(ln k)-0.5. Two formulas which have a very close behaviour.