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.