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Conference Papers Year : 2018

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

Abstract

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.
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Dates and versions

hal-02199638 , version 1 (09-12-2019)

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  • HAL Id : hal-02199638 , version 1

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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⟩

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