WHAT IS a sandpile?

An abelian sandpile is a collection of indistinguishable chips distributed among the vertices of a graph. More precisely, it is a function from the vertices to the nonnegative integers, indicating how many chips are at each vertex. A vertex is called unstable if it has at least as many chips as its degree, and an unstable vertex can topple by sending one chip to each neighboring vertex. Note that toppling one vertex may cause neighboring vertices to become unstable. If the graph is connected and infinite, and the number of chips is finite, then all vertices become stable after finitely many topplings. An easy lemma says that the final stable configuration is independent of the order of topplings (this is the reason for calling sandpiles “abelian”). For instance, start with a large pile of chips at the origin of the square grid Z2 and perform topplings until every vertex is stable. The process gives rise to a beautiful large-scale pattern (Figure 1). More generally, one obtains different patterns by starting with a constant number h ≤ 2d−2 of chips at each site in Zd and adding n chips at the origin; see Figure 3 for two examples. Sandpile dynamics have been invented numerous times, attached to such names as chipfiring, the probabilistic abacus, and the dollar game. The name “sandpile” comes from statistical physics, where the model was proposed in a famous 1987 paper of Bak, Tang and Wiesenfeld as an example of self-organized criticality, or the tendency of physical systems to drive themselves toward critical, barely stable states. In the original BTW model, chips are added at random vertices of an N ×N box in Z2. Each time a chip is added, it may cause an avalanche of topplings. If this avalanche reaches the boundary, then topplings at the boundary cause chips to disappear from the system. In the stationary state, the distribution of avalanche sizes has a power-law tail: very large avalanches occur quite frequently (e.g., the expected number of topplings in an avalanche goes to infinity with N). To any finite connected graph G we can associate an abelian group K(G), called the sandpile group. This group is an isomorphism invariant of the graph and reflects certain combinatorial information about the graph. To define the group, we single out one vertex of G as the sink and ignore chips that fall into the sink. The operation of addition followed by stabilization gives the set M of all stable sandpiles on G the structure of a commutative monoid. An ideal of M is a subset J ⊂M satisfying σJ ⊂ J for all σ ∈M . The sandpile group K(G) is the minimal ideal of M (i.e., the intersection of all ideals). The minimal ideal of a finite commutative monoid is always a group. (We encourage readers unfamiliar with this remarkable fact to prove it for themselves.) One interesting feature of constructing a group in this manner is that it is not at all obvious what the identity element is! Indeed, for many graphs G the identity element of K(G) is a highly nontrivial object with intricate structure (Figure 2). To realize the sandpile group in a more concrete way, we can view sandpiles σ as elements of the free abelian group ZV , where V is the set of non-sink vertices of G. Toppling a vertex v corresponds to adding the vector ∆v to σ, where