Recurrence of horizontal-vertical walks

Abstract

Consider a nearest neighbor random walk on the two-dimensional integer lattice, where each vertex is initially labeled either `H' or `V', uniformly and independently. At each discrete time step, the walker resamples the label at its current location (changing `H' to `V' and `V' to `H' with probability $q$). Then, it takes a mean zero horizontal step if the new label is `H', and a mean zero vertical step if the new label is `V'. This model is a randomized version of the deterministic rotor walk, for which its recurrence (i.e., visiting every vertex infinitely often with probability 1) in two dimensions is still an open problem. We answer the analogous question for the the horizontal-vertical walk, by showing that the horizontal-vertical walk is recurrent for $q\in (\frac{1}{3},1]$.

33 pages, 3 figures. Extra references added. To appear in Ann. Inst. Henri Poincaré Probab. Stat

References in corpus (7)

• A survey of random processes with reinforcement
• Recurrence of edge-reinforced random walk on a two-dimensional graph
• Rotor walks on transient graphs and the wired spanning forest
• Abelian networks IV. Dynamics of nonhalting networks
• Solution manifold and Its Statistical Applications
• Random walks with local memory
• A rotor configuration with maximum escape rate

Cited by in corpus (1)

• Random walks with local memory