Discrete Mathematics : The Abelian Sandpile Model

Decomposing recurrent states of the abelian sandpile model

The recurrent states of the Abelian sandpile model (ASM) are those states that appear infinitely often. For this reason they occupy a central position in ASM research. We present several new results for classifying recurrent states of the Abelian sandpile model on graphs that may be decomposed in a variety of ways. These results allow us to classify, for certain families of graphs, recurrent st...

Concrete and abstract structure of the sandpile group for thick trees with loops

We answer a question of László Babai concerning the abelian sandpile model. Given a graph, the model yields a finite abelian group of recurrent configurations which is closely related to the combinatorial Laplacian of the graph. We explicitly describe the group elements and operations in the case of thick trees with loops—that is, graphs which are obtained from trees by setting arbitrary edge m...

The Gergonne p-pile Problem and the Dynamics of the Function x | => (x + r)/p

The Abelian Sandpile and Related Models

The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality. This paper presents a brief review of known results about the model. The abelian group structure of the algebra of operators allows an exact calculation of many of its properties. In particular, when there is a preferred direction, one can calculate all the critical exponents characterizing th...

Laplacian Growth, Sandpiles and Scaling Limits

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 survey progress over the last decade on discrete models of (internal) Laplacian growth, including the abelian sandpile, internal DLA, rotor aggregation, and the scaling limits of these models on the lattice Z as the ...