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

TL;DR

This paper investigates the capacity of the range of simple random walks on three and four-dimensional integer lattices, establishing a law of large numbers in four dimensions and distributional convergence in three dimensions.

Contribution

It provides the first weak law of large numbers for the capacity in four dimensions and shows convergence in distribution to Brownian motion in three dimensions.

Findings

Weak law of large numbers for capacity on $\

convergence in distribution to Brownian motion in 3D

new insights into the asymptotic behavior of random walk capacity

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.