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

TL;DR

This paper investigates the capacity of the range of a transient simple random walk on integer lattices, establishing a central limit theorem for dimensions six and higher, and discusses open questions for lower dimensions.

Contribution

It proves a central limit theorem for the capacity of the range of random walks in high dimensions ($d extgreater=6$), advancing understanding of their probabilistic behavior.

Findings

Central limit theorem established for $d extgreater=6$

Analysis of capacity behavior in high-dimensional random walks

Open questions posed for lower dimensions

Abstract

We study the capacity of the range of a transient simple random walk on $\mathbb{Z}^d$. Our main result is a central limit theorem for the capacity of the range for $d\ge 6$. We present a few open questions in lower dimensions.