Expected Number of Distinct Sites Visited by a Random Walk with an Infinite Variance

Consider a random walk of n steps on an infinite, simple cubic lattice. Let p(r) be the (symmetric) probability of a vector jump r, and let Sn be the expected number of distinct lattice points visited in the course of the random walk. In the present paper we calculate asymptotic values for Sn for the particular choice of jump probabilities p(r) = p(−r) = Ar−(1+α), where 2 ≥ α > 0, and p(r1, r2) = Ar−β, where r2=r12 + r22, 2≥β >1, and A denotes the normalizing constant. The results are, in 1D, (1) Sn ∼ An, 1 > α > 0, (2) Sn ∼ Bn/ln n, α = 1, (3) Sn ∼ Cn1/α, 2 > α > 1, (4) Sn ∼ D(n ln n)½, α = 2, where A, B, C, and D are calculable constants, and, in 2D, (1) Sn ∼ An, 2 > β > 1, (2) Sn ∼ Bn/ln ln n, β = 2.