Capacity of the range in dimension $5$

Abstract

We prove a central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb{Z}^{5}$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular, an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covariances of some intersection events at the leading order.