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arXiv · 2610.32687

Avalanches of the Abelian sandpile on the hierarchical lattice

Luca Cattani·Leandro Chiarini·Alessandra Cipriani
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Abstract

We study the Abelian sandpile model on a sequence of finite fractal graphs: the hierarchical, or Migdal--Kadanoff, lattices. Our first main result consists of matching power-law tail bounds for the avalanche and the uniform wave started at certain "pivotal" vertices. We also establish a limiting result for the pivotal reach of the avalanche and, finally, a scaling limit for the size of a uniform wave started at a vertex of high degree.

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Luca Cattani, Leandro Chiarini, Alessandra Cipriani. 2026-10-23. Avalanches of the Abelian sandpile on the hierarchical lattice. https://arxiv.org/abs/2610.32687

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