Extreme slowdowns for one-dimensional excited random walks

We study the asymptotics of the probabilities of extreme slowdown events for transient one-dimensional excited random walks. That is, if $\{X_n\}_{n\geq 0}$ is a transient one-dimensional excited random walk and $T_n = \min\{ k: \, X_k = n\}$, we study the asymptotics of probabilities of the form $P(X_n \leq n^\gamma)$ and $P(T_{n^\gamma} \geq n )$ with $\gamma < 1$. We show that there is an interesting change in the rate of decay of these extreme slowdown probabilities when $\gamma < 1/2$.