Capacity of the range of random walk on $\mathbb{Z}^{4}$

Abstract

We study the scaling limit of the capacity of the range of a random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-Gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in ’86 [Comm. Math. Phys. 104 (1986) 471–507] for the volume of the range in dimension two.