Deviations for the Capacity of the Range of a Random Walk

TL;DR

This paper provides estimates for large and moderate deviations of the capacity of a random walk's range in high dimensions, extending previous work on the volume of the range with adapted strategies.

Contribution

It introduces new deviation estimates for the capacity of the range of a random walk in dimensions d≥5, adapting methods from prior volume range studies.

Findings

Deviation estimates for capacity in high dimensions

Analogous results to volume range deviations

Methodology adaptable from previous volume studies

Abstract

We obtain estimates for large and moderate deviations for the capacity of the range of a random walk on $\mathbb{Z}^d$, in dimension $d\ge 5$, both in the upward and downward directions. The results are analogous to those we obtained for the volume of the range in two companion papers [AS17, AS19]. Interestingly, the main steps of the strategy we developed for the latter apply in this seemingly different setting, yet the details of the analysis are different