The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing th...

We consider the directed Abelian sandpile model in the presence of sink sites whose density ft at depth t below the top surface varies as c t . For χ > 1 the disorder is irrelevant. For χ < 1, it is relevant and the model is no longer critical for any nonzero c. For χ = 1 the exponents of the avalanche distributions depend continuously on the amplitude c of the disorder. We calculate this depen...

Hereditary chip-firing models generalize the Abelian sandpile model and the cluster firing model to an exponential family of games induced by covers of the vertex set. This generalization retains some desirable properties, e.g. stabilization is independent of firings chosen and each chip-firing equivalence class contains a unique recurrent configuration. In this paper we present an explicit bij...

This is a draft of a primer on the algebraic geometry of the Abelian Sandpile Model. version:July, 11, 2009.

Many invertible actions τ on a set S of combinatorial objects, along with a natural statistic f on S, exhibit the following property which we dub homomesy: the average of f over each τ -orbit in S is the same as the average of f over the whole set S. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong,...

In this paper we study the identity of the Abelian Sandpile Model on a rectangular lattice. This configuration can be computed with the burning algorithm, which, starting from the empty lattice, computes a sequence of configurations, the last of which is the identity. We extend this algorithm to an infinite lattice, which allows us to prove that the first steps of the algorithm on a finite latt...

Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to ...

Following the lead of earlier researchers, we construct a real space renormalization group map for a class of self-organized critical models. We then test a series of quantitative predictions implied by this map against direct simulations of various models. The theory passes four tests, but apparently fails a fifth. We also find that the map has an interesting nongeneric property, suggesting a ...

Given an undirected graph G = (V, E), and a designated vertex q ∈ V , the notion of a G-parking function (with respect to q) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of maximum G-parking functions a...

We present a transfer matrix method which is particularly useful for solving some classes of sandpile models. The method is then used to solve the de-terministic nonabelian sandpile models for N=2 and N=3. The possibility of generalization to arbitrary N is discussed briefly.