Large deviations and slowdown asymptotics for one - dimensional excited random walks ∗

We study the large deviations of excited random walks on 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 v0, then the large deviation rate function for the position of the excited random walk is zero on the interval [0, v0] and so probabilities such as P (Xn < nv) for v ∈ (0, v0) decay subexponentially. We show that rate of decay for such slowdown probabilities is polynomial of the order n1−δ/2, where δ > 2 is the expected total drift per site of the cookie environment.