Two observations on the capacity of the range of simple random walks on ${\mathbb{Z}}^3$ and ${\mathbb{Z}}^4$

Abstract

We prove a weak law of large numbers for the capacity of the range of simple random walks on ${\mathbb{Z}}^4$. On ${\mathbb{Z}}^3$, we show that the capacity, properly scaled, converges in distribution towards the corresponding quantity for three dimensional Brownian motion.