Excursions and occupation times of critical excited random walks

TL;DR

This paper investigates the behavior of critical excited random walks on integers, focusing on excursions, occupation times, and scaling limits at the critical transition point where the expected drift equals one.

Contribution

It provides new estimates and analyses of excursions, occupation times, and scaling limits specifically for the critical case of excited random walks, addressing challenges where previous methods fail.

Findings

Excursions from the origin have specific depth and duration characteristics at criticality.

Occupation times of positive and negative axes converge to beta distributions in the non-critical recurrent case.

Scaling limits of ERWs are characterized at the critical transition point.

Abstract

The paper considers excited random walks (ERWs) on integers in i.i.d. environments with a bounded number of excitations per site. The emphasis is primarily on the critical case for the transition between recurrence and transience which occurs when the total expected drift $\delta$ at each site of the environment is equal to 1 in absolute value. Several crucial estimates for ERWs fail in the critical case and require a separate treatment. The main results discuss the depth and duration of excursions from the origin for $|\delta|=1$ as well as occupation times of negative and positive semi-axes and scaling limits of ERW indexed by these occupation times. It is also pointed out that the limiting proportions of the time spent by a non-critical recurrent ERW (i.e. when $|\delta|<1$) above or below zero converge to beta random variables with explicit parameters given in terms of $\delta$.