Cutoff and mixing time for transient random walks in random environments

We show that a sequence of birth-and-death chains, given by lazy random walks in a (transient) environment (RWRE) on [0; n], exhibits a cutoff in the ballistic regime but does not exhibit a cutoff in the (interior of) the subballistic regime. We investigate the growth of the mixing times for this model. As an important step in the proof, we derive bounds for the quenched expectation and the quenched variance of the hitting times of the RWRE, which are of independent interest.