On the Strong Recurrence of Recurrent RWRE

We consider a recurrent RWRE $(X_n{)}_{n\in {\mathbb{N}}_0}$ on $\mathbb{Z}$ and investigate the quenched return probabilities of the RWRE to the origin for which we state results on their decay in terms of summability. Additionally, we give some examples for recurrent RWRE with a multidimensional state space which give reason for the part “strong recurrence” in the title of this paper when we compare the behaviour of RWRE with the behaviour of the symmetric random walk on ${\mathbb{Z}}^d$.