Random walks on oriented lattices

We consider simple random walks on two partially directed lattices. Though superficially similar, they exhibit different recurrence/transience behaviors. Our main result is indeed a proof of recurrence for one of the graphs, solving a conjecture of Menshikov et al. ('17). For the other one, we analyze the asymptotics of the return probabilities, providing a new proof of its transience. Furthermore, we study the limiting laws of the winding number around the origin for these walks. This is joint work with Gianluca Bosi and Yuval Peres.