# Capacity of the range in dimension 5

## TL;DR

This paper establishes a central limit theorem for the capacity of the range of symmetric random walks in five dimensions, revealing a logarithmic correction in variance scaling and using new intersection probability estimates.

## Contribution

It provides the first CLT for the capacity of the range in dimension five, with novel asymptotic estimates for intersection probabilities in higher dimensions.

## Key findings

- Central limit theorem for capacity in dimension 5
- Logarithmic correction in variance scaling
- New asymptotic estimates for intersection probabilities

## Abstract

We prove a Central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb Z^5$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge 5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covariances of some intersection events, at the leading order.