Decomposing Recurrent States of the Abelian Sandpile Model

Mark Dukes ; Thomas Selig

Séminaire lotharingien de combinatoire, Tome 77 (2017-2018)

Résumé

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. The set of stable configurations on a graph form a Markov chain whereby a transition from a configuration c to another c' occurs if the addition of a grain to c and the resulting sequence of topplings (if any) yields the state c'. Checking whether a stable configuration is recurrent is a far from trivial task and requires Dhar's criterion, an algorithmic process, to be used. 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 represent an enormous computational saving with respect to Dhar's criterion. Furthermore, they allow us to classify, for certain families of graphs, recurrent states in terms of the recurrent states of its components. We use these decompositions to give recurrence relations for the generating functions of the level statistic on the recurrent configurations. We also interpret our results with respect to the sandpile group.