Deviations for the Capacity of the Range of a Random Walk

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

This version covers the whole moderate deviation regime in $d\ge 7$