The University of Chicago on the Abelian Sandpile Model a Dissertation Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy Department of Mathematics by Evelin Christiana Toumpakari

The Abelian Sandpile Model is a diffusion process on graphs, studied, under various names, in statistical physics, theoretical computer science, and algebraic graph theory. The model takes a rooted directed multigraph X , the ambient space, in which the root is accessible from every vertex, and associates with it a commutative monoidM, a commutative semigroup S, and an abelian group G as follows. For vertices i, j, let aij denote the number of i→ j edges and let deg(i) denote the out-degree of i. Let V be the set of ordinary (non-root) vertices. With each i ∈ V associate a symbol xi and consider the relations deg(i)xi = ∑ j∈V aijxj . LetM, S, and G be the commutative monoid, semigroup and group, respectively, generated by {xi : i ∈ V } subject to these defining relations. M is the sandpile monoid, S is the sandpile semigroup, and G is the sandpile group associated with X . We write the operation additively, so 0 is the identity of M. We have M = S ∪ {0}; we show that G is the unique minimal ideal of M. The main results of the thesis cover two areas: (1) a general study of the structure of the sandpile monoid and (2) detailed analysis of the structure of the sandpile group for a special class of graphs. Our first main goal is to establish connections between the algebraic structure of M, S, G, and the combinatorial structure of the underlying ambient space X . M turns out to be a distributive lattice of semigroups each of which has a unique idempotent. The distributive lattice in question is the lattice L of idempotents ofM; L turns out to be isomorphic to the dual of the lattice of ideals of the poset of normal strong components of X . The M→ L epimorphism defines the smallest semilattice congruence of M; therefore L is the universal semilattice of M. We characterize the directed graphs X for which S has a unique idempotent; this includes the important case when the digraph induced on the ordinary vertices is strongly connected. If the idempotent in S is unique then the Rees quotient S/G