The Range of a Rotor Walk

Abstract

In rotor walk on a graph, the exits from each vertex follow a prescribed periodic sequence. We show that any rotor walk on the d-dimensional lattice ℤd visits at least on the order of td/(d+1) distinct sites in t steps. This result extends to Eulerian graphs with a volume growth condition. In a uniform rotor walk, the first exit from each vertex is to a neighbor chosen uniformly at random. We prove a shape theorem for the uniform rotor walk on the comb graph, showing that the size of the range is of order t2/3 and the asymptotic shape of the range is a diamond. Using a connection to the mirror model, we show that the uniform rotor walk 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.