Excursions and Occupation times of Critical Excited Random Walks

We consider 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 δ 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 |δ| = 1 as well as occupation times of negative and positive semiaxes and scaling limits of ERW indexed by these occupation times. We also point out that the limiting proportions of the time spent by a non-critical recurrent ERW (i.e. when |δ| < 1) above or below zero converge to beta random variables with explicit parameters given in terms of δ. The last observation can be interpreted as an ERW analog of the arcsine law for the simple symmetric random walk.