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The sandpile model is a discrete event dynamical system always considered on a fixed graph with certain toppling rules for the stabilization of the configurations of the graph. In this work we first consider the Abelian sandpile on a 2-regular graph: show the structure of the recurrent configurations, idempotent, and generators of the recurrent group, and conclude it is cyclic. Second, we consider the non-Abelian aperiodic sandpile semigroup on a directed rooted tree: show the aperiodic complexity of the semigroup and constructively show the emulation of the sandpile semigroup by a minimum length wreath product of irreducible blocks.

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The Study of the Transformation Semigroup of the Abelian and Directed Non-Abelian Sandpiles

  • 会议论文

  • 发表日期：

    2025年

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  • Hanna Derets,
  • Chrystopher L. Nehaniv

摘要

The sandpile model is a discrete event dynamical system always considered on a fixed graph with certain toppling rules for the stabilization of the configurations of the graph. In this work we first consider the Abelian sandpile on a 2-regular graph: show the structure of the recurrent configurations, idempotent, and generators of the recurrent group, and conclude it is cyclic. Second, we consider the non-Abelian aperiodic sandpile semigroup on a directed rooted tree: show the aperiodic complexity of the semigroup and constructively show the emulation of the sandpile semigroup by a minimum length wreath product of irreducible blocks.