Capacity of the range of random walk: Moderate deviations in dimensions 4 and 5

TL;DR

This paper establishes a moderate deviation principle for the capacity of the range of a random walk in five-dimensional integer lattice, revealing different tail behaviors depending on the deviation scale, and extends previous results to a broader range.

Contribution

It proves a comprehensive moderate deviation principle for the capacity of random walk ranges in dimensions 4 and 5, including new tail behavior regimes and extending prior work.

Findings

Gaussian tails for small deviations

Non-Gaussian tails for larger deviations

Extension of deviation results to dimension 4

Abstract

We prove a moderate deviation principle for the capacity of the range of random walk in $\mathbb{Z}^5$. Depending on the scale of deviation, we get two different regimes. We observe Gaussian tails when the deviation scale is smaller than $n^{1/2} (\log n)^{3/4}$. Otherwise, we get non-Gaussian tails with a constant arising from a generalized Gagliardo-Nirenberg inequality. This is analogous to the behavior of the volume of the random walk range in $\mathbb{Z}^3$. Our methods can also be applied to the $d = 4$ case to prove the moderate deviation principle in almost the full range of interest. This extends the work of Okada and the first author \cite{AdhikariOkada2023}, where they showed moderate deviations up to a deviation scale of $\log \log n$ times the standard deviation.