Excursions and occupation times of critical excited random walks

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 $\delta$.

23 pages; minor changes, added Appendix B with some technical details