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

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.

36 pages. Minor revisions