# Student Probability Seminar

#### The Range of Rotor Walk and Recurrence of Directed Lattices

**Speaker:**
Laura Florescu

**Location:**
Warren Weaver Hall 905

**Date:**
Tuesday, November 26, 2013, 3:30 p.m.

**Synopsis:**

Rotor walk is a deterministic analogue of random walk introduced by Jim Propp. In a rotor walk on a graph, a particle exits a vertex in a predetermined cyclic fashion. We show a lower bound of \(t^{d/d+1}\) on the number of sites visited by iid rotor walk in \(t\) steps on any lattice in a \(d\) dimensional space, thus proving the lower bound of \(t^{2/3}\) conjectured for the square grid 17 years ago. Additionally, we show that the range of rotor walk on the comb lattice is also \(t^{2/3}\), contrasting with that of random walk which is \(\sqrt{t}\log{t}\). We also show recurrence of rotor walk on the Manhattan and F-lattices through connections to percolation arising from the 'stochastic pin ball' also known as the Lorentz wind-tree mirror model. This is recent work with Lionel Levine and Yuval Peres.