2014/10/26 by Dmitry Dolgopyat, Dolgopyat, Dmitry, Elena Kosygina +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60F17 #60J80. Secondary: 60J60 #Diffusion and Search Dynamics #FOS: Mathematics #Primary: 60K37 #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1410.7090
openalex publication_date 2014/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 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 |δ|<1) above or below zero converge to beta random variables with explicit parameters given in terms of δ.