The Mathematics of Chip-firing is a solid introduction and overview of the growing field of
chip-firing. It offers an appreciation for the richness and diversity of the subject. Chip-firing …

We study the scaling limits of three different aggregation models on ℤ d: internal DLA, in
which particles perform random walks until reaching an unoccupied site; the rotor-router …

Activated Random Walks, on Z d for any d≥ 1, is an interacting particle system, where
particles can be in either of two states: active or frozen. Each active particle performs a …

We study the abelian sandpile growth model, where n particles are added at the origin on a
stable background configuration in ℤ d. Any site with at least 2 d particles then topples by …

The rotor walk is a derandomized version of the random walk on a graph. On successive
visits to any given vertex, the walker is routed to each of the neighboring vertices in some …

Let each of $ n $ particles starting at the origin in $\mathbb Z^ 2$ perform simple random
walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that …

The Abelian sandpile process evolves configurations of chips on the integer lattice by
toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 …

The Abelian sandpile growth model is a diffusion process for configurations of chips placed
on vertices of the integer lattice Z d, in which sites with at least 2 d chips topple, distributing …

In Deepak Dhar's model of abelian distributed processors, automata occupy the vertices of a
graph and communicate via the edges. We show that two simple axioms ensure that the final …

This survey is an extended version of lectures given at the Cornell Probability Summer
School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and …