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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.

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