Absorption time by a random trap distribution

Consider a particle performing a random walk to nearest neighbors on a simple cubic lattice in any number of dimensions D. The lattice contains a fraction q of randomly located ’’trapping’’ sites which absorb the walker when stepped on. We calculate the mean time to trapping. This involves the expected number V(t) of distinct sites that a walker would visit in t steps in absence of traps, a quantity known only asymptotically for large t; however, an exact calculation possible in one dimension suggests that the results are unexpectedly precise. The trapping time is proportional to q−1 for D=3 or greater, and more complicated for D=1,2.