Primer for the algebraic geometry of sandpiles

The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the Laplacian matrix, drawing out connections with the ASM. An extended summary of the ASM and of the required algebraic geometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new construction of arithmetically Gorenstein ideals; a generalization to directed multigraphs of a duality theorem between elements of the sandpile group of a graph and the graph's superstable configurations (parking functions); and a characterization of the top Betti number of the minimal free resolution of the Laplacian lattice ideal as the number of elements of the sandpile group of least degree. A characterization of all the Betti numbers is conjectured.Comment: 45 pages, 14 figures. v2: corrected typo

The Abelian Sandpile Model on an Infinite Tree

We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality

Determining Genus From Sandpile Torsor Algorithms

We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus.Comment: Extended Abstract Accepted to FPSAC 2018. Revision of previous versio

Sandpiles, spanning trees, and plane duality

Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.Comment: 13 pages, 9 figure