A definition and counting of biperiodic recurrent configurations in the sandpile model on square lattice

Yvan Le Borgne (1) , Henri Derycke (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

For the sandpile model on the usual two dimensional grid, we propose a weaker version of Dhar criterion to define recurrent configurations among stable biperiodic configurations. We check this new criterion via an algorithm which auto-stabilises to a canonical ultimately periodic behaviour independent of details in its not fully specified initialisation. This leads to ultimately periodic edge/vertex traversals similar to those of Cori-Le Borgne [2] in the case of finite graphs and then to a bijection with some cycle-rooted forests on the torus describing the period. A determinantal formula [5] counts all those forests and the refinement with some monodromy parameters allows to identify in some coefficients the number of recurrent configurations.

Domaines: Combinatoire [math.CO]

Yvan Le Borgne, Henri Derycke. A definition and counting of biperiodic recurrent configurations in the sandpile model on square lattice. 17e journées montoises d'informatique théorique, Sep 2018, Bordeaux, France. ⟨hal-02199638⟩