Capacity of the range of random walk: Moderate deviations in dimensions 4 and 5
Arka Adhikari, Jiyun Park
TL;DR
The paper proves a nuanced moderate deviation principle for the capacity of the random walk range in $\mathbb{Z}^5$, revealing a Gaussian regime for small deviation scales and a non-Gaussian regime governed by a generalized Gagliardo–Nirenberg inequality for larger scales, with a precise transition at $b_n \asymp \sqrt{\log n}$. A novel graph-based representation of high-moment interactions is developed to control singular dependencies and to enable induction across many configurations, which is central to capturing the cross-term dynamics. The results extend to $d=4$ and improve prior work by establishing the Gaussian regime in dimension five and tightening constants, linking capacity deviations to a Brownian-graph functional via a Gartner–Ellis framework. The findings illuminate deep connections between capacity fluctuations, intersection structures, and functional-analytic inequalities, with potential applications to branching random walks and diagrammatic methods in probability and statistical physics.
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.