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 …

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 …

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 …

We prove that the set of quadratic growths attainable by integer-valued superharmonic
functions on the lattice Z^2 has the structure of an Apollonian circle packing. This completely …

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 …

Laplacian growth is the study of interfaces that move in proportion to harmonic measure.
Physically, it arises in fluid flow and electrical problems involving a moving boundary. We …

Tropical geometry, an established field in pure mathematics, is a place where string theory,
mirror symmetry, computational algebra, auction theory, and so forth meet and influence one …

We show that the patterns in the Abelian sandpile are stable. The proof combines the
structure theory for the patterns with the regularity machinery for non-divergence form elliptic …

For a real projective variety X, the cone \Sigma_X of sums of squares of linear forms plays a
fundamental role in real algebraic geometry. The dual cone \Sigma_X^* is a spectrahedron …

Abstract Activated Random Walk is a particle system displaying Self-Organized Criticality, in
that the dynamics spontaneously drive the system to a critical state. How universal is this …