Revolving random walks in 2D: analysis of type and winding number.

We consider simple random walks on two directed versions of the $\mathbb{Z}^2$ lattice; one characteristic feature of these random walks is that they are bound to revolve, according to the orientation prescribed by the edges. The first model was studied by Campanino and Petritis(‘03) and shown to be transient; the other one appeared recently in a paper by Menshikov et al. (’17), where the authors conjectured its recurrence. We shall indeed confirm this conjecture: our proof is done by considering a continuous analogue of the random walk and then applying the Lyapunov function criteria. On the other hand, we obtain a local limit theorem for the return probabilities of the first random walk, which in particular gives a new proof of transience. Finally, we deduce some results related to the winding number for both random walks. This results are joint work with Y.Peres (Microsoft Research, Redmond) and Y.Hu (University of Washington).