The range of a rotor walk

TL;DR

This paper investigates the behavior of rotor walks on various graphs, establishing lower bounds on visited sites, a shape theorem for the comb graph, and recurrence properties on certain directed lattices.

Contribution

It provides new bounds on the range of rotor walks, proves a shape theorem for the comb graph, and demonstrates recurrence on specific directed lattices.

Findings

Rotor walk visits at least on the order of t^{d/(d+1)} sites on Eulerian graphs.

Range on the comb graph is of order t^{2/3} with a diamond shape.

Rotor walk is recurrent on certain directed lattices.

Abstract

In a \emph{rotor walk} the exits from each vertex follow a prescribed periodic sequence. On an infinite Eulerian graph embedded periodically in $\R^d$, we show that any simple rotor walk, regardless of rotor mechanism or initial rotor configuration, visits at least on the order of $t^{d/(d+1)}$ distinct sites in $t$ steps. We prove a shape theorem for the rotor walk on the comb graph with i.i.d.\ uniform initial rotors, showing that the range is of order $t^{2/3}$ and the asymptotic shape of the range is a diamond. Using a connection to the mirror model and critical percolation, we show that rotor walk with i.i.d.\ uniform initial rotors is recurrent on two different directed graphs obtained by orienting the edges of the square grid, the Manhattan lattice and the $F$-lattice. We end with a short discussion of the time it takes for rotor walk to cover a finite Eulerian graph.