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 {Xn}n≥0{Xn}n≥0 is a transient one-dimensional excited random walk and Tn=min{k:Xk=n}, we study the asymptotics of probabilities of the form P(Xn≤nγ)P(Xn≤nγ) and P(Tnγ≥n)P(Tnγ≥n) with γ<1γ<1. We show that there is an interesting change in the rate of decay of these extreme slowdown probabilities when γ<1/2γ<1/2.