Escape Rates for Rotor Walks in $\mathbb{Z}^d$

Abstract

Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on ${\mathbb Z}^d$ for the initial configuration of all rotors aligned. If $n$ particles in turn perform rotor walks starting from the origin, we show that the number that escape (i.e., never return to the origin) is of order $n$ in dimensions $d \geq 3$ and of order $n/\log n$ in dimension $2$.