Number of distinct sites visited by a random walker in the presence of a trap

Abstract

The authors study the number of distinct sites visited by a random walker in d=1 after t steps, S(t), in the presence of a trap. They calculate the distribution q(S, t) of S(t) in the limit of large t. They find an unusual crossover in the probability density at S approximately=Sx identical to Dt. For S<<Sx, q(S, t) approximately S-2 and for S>>Sx, q(S, t) approximately St-3/2 exp(-S2/4Dt). Fro this crossover it follows that the mean number of distinct sites visited is (S(t)) approximately In(t).