Abstract
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 $P(X_n 2$ is the expected total drift per site of the cookie environment.
Highlights
In this paper we study the large deviations for one-dimensional excited random walks
We are primarily concerned with the large deviations of excited random walks
In a similar manner to the approach used for large deviations of random walks in random environments, we deduce a large deviation principle for Xn/n from a large deviation principle for Tn/n, where
Summary
In this paper we study the large deviations for one-dimensional excited random walks. When the walker visits the site x for the i-th time, he eats the i-th cookie which causes his step to be as a simple random walk with parameter ωx(i) For this reason we will refer to ω = {ωi(j)}i∈Z, j≥1 as a cookie environment. The model of excited random walks was further generalized by Zerner and Kosygina to allow for cookies with both positive and negative drifts [14]. Basdevant and Singh solved this problem in [1] where they showed that v0 > 0 if and only if δ > 2 These results for recurrence/transience and the limiting speed were given only for cookies with non-negative drift but were recently generalized by Kosygina and Zerner [14] to the general model we described above that allows for cookies with both positive and negative drifts. The interested reader is referred to the papers [2, 10, 11, 13, 14] for more information on limiting distributions
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