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Article Dans Une Revue Discrete Mathematics Année : 2002

On the Identity of the Sandpile Group

Résumé

In 1991, Dhar \cite{Dhar},\cite{DharRuelle} proves that the recurrent configurations of the sandpile automaton form an abelian group for the addition operator $\oplus$. In this article we study the identity element of this group for the sandpile automaton on rectangular grids of size $p \times q$. We prove that for $q \geq p(2+3 \sqrt 2)/2$, this identity is made of $3$ parts ($x < \frac{p(2+3\sqrt{2})}{4}$,$\frac{p(2+3\sqrt{2})}{4} < x < p - \frac{p(2+3\sqrt{2})}{4}$, $x > p - \frac{p(2+3\sqrt{2})}{4}$.) Extremal parts are symmetric whereas the central one has $2$ grains of sand on every vertex. We give a new method to compute the identity element of the group. This method is twice as fast experimentally as the other known methods.
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Dates et versions

hal-00016377 , version 1 (02-01-2006)

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  • HAL Id : hal-00016377 , version 1

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Dominique Rossin, Yvan Le Borgne. On the Identity of the Sandpile Group. Discrete Mathematics, 2002, 256, 3, pp.775--790. ⟨hal-00016377⟩
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