Capacity of the range of random walk on $\mathbb{Z}^4$

TL;DR

This paper investigates the asymptotic behavior of the capacity of a simple random walk's range on , establishing laws of large numbers and a non-Gaussian central limit theorem, analogous to known results for volume in .

Contribution

It provides the first detailed analysis of the capacity's scaling limit in four dimensions, including a strong law and a non-Gaussian CLT.

Findings

Established a strong law of large numbers for capacity in 

Proved a non-Gaussian central limit theorem for the capacity

Identified the asymptotic behavior analogous to the volume in 

Abstract

We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 for the volume of the range in dimension two.