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

TL;DR

This paper investigates the large deviations of one-dimensional excited random walks, providing a large deviation principle for hitting times and positions, and characterizing the decay rates of slowdown probabilities.

Contribution

It establishes a large deviation principle for excited random walks and describes the polynomial decay of slowdown probabilities in the transient case with positive speed.

Findings

Large deviation principle for hitting times and positions.

Rate function is zero on [0, v_0], indicating subexponential decay.

Decay rate for slowdown probabilities is polynomial, specifically n^{1 - δ/2}.

Abstract

We study the large deviations of one-dimensional excited random walks. 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 $P(X_n < nv)$ 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.