Large deviations and slowdown asymptotics for one-dimensional excited random walks. (English) Zbl 1260.60184

Peterson, Jonathon

Electron. J. Probab. 17, Paper No. 48, 24 p. (2012).

Summary: We study the large deviations of excited random walks on \(\mathbb{Z}\). We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed \(v_0\), then the large deviation rate function for the position of the excited random walk is zero on the interval \([0,v_0]\) and so probabilities such as \(\operatorname{P}(X_n < nv)\) for \(v \in (0,v_0)\) decay subexponentially. We show that rate of decay for such slowdown probabilities is polynomial of the order \(n^{1-\delta/2}\), where \(\delta>2\) is the expected total drift per site of the cookie environment.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F10	Large deviations
60K37	Processes in random environments

Keywords:

excited random walk; large deviations