Lionel Levine

Chip-Firing and Rotor-Routing on Directed Graphs

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work (Levine and Peres, Indiana Univ Math J 57(1):431–450, 2008). For the shape consisting of

Scaling limits for internal aggregation models with multiple sources

n=\omega_d r^d

Spherical asymptotics for the rotor-router model in Zd

sites, where ωd is the volume of the unit ball in

Sandpile groups and spanning trees of directed line graphs

\mathbb{R}^d

Logarithmic Fluctuations for Internal DLA

, we show that the inradius of the set of occupied sites is at least r − O(logr), while the outradius is at most r + O(rα) for any α > 1 − 1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with n = πr2 particles, we show that the inradius is at least

The rotor-router shape is spherical

r/\sqrt{3}

The sandpile group of a tree

, and the outradius is at most

The looping constant of Z d

(r+o(r))/\sqrt{2}

Driving sandpiles to criticality and beyond.

. This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig.