  This work is motivated by the study of some two-dimensional random walks in
random environment (RWRE) with transition probabilities independent of one
coordinate of the walk. These are non-reversible models and can not be treated
by electrical network techniques. The proof of the recurrence of such RWRE
needs new estimates for quenched return probabilities of a one-dimensional
recurrent RWRE. We obtained these estimates by constructing suitable valleys
for the potential. They imply that k independent walkers in the same
one-dimensional (recurrent) environment will meet in the origin infinitely
often, for any k. We also consider direct products of one-dimensional recurrent
RWRE with another RWRE or with a RW. We point out the that models involving
one-dimensional recurrent RWRE are more recurrent than the corresponding models
involving simple symmetric walk.
