Recurrence of horizontal-vertical walks

TL;DR

This paper studies a randomized 2D lattice walk with label-dependent directions, showing recurrence for certain resampling probabilities, advancing understanding of related deterministic rotor walks whose recurrence remains unresolved.

Contribution

It proves recurrence of the horizontal-vertical walk for resampling probability q in (1/3, 1], providing insights into the behavior of similar deterministic models.

Findings

The walk is recurrent for q in (1/3, 1].

Recurrence holds with probability 1 for these parameters.

The result advances understanding of rotor walk behavior.

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 $q \in (\frac{1}{3},1]$.