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PreprintarXiv licencearXiv

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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Cite

BibTeXRIS
BibTeX
@misc{cattani2026avalanches,
  title = {{Avalanches of the Abelian sandpile on the hierarchical lattice}},
  author = {Luca Cattani and Leandro Chiarini and Alessandra Cipriani},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32687},
  eprint = {2610.32687},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32687},
}
RIS
TY  - UNPB
TI  - Avalanches of the Abelian sandpile on the hierarchical lattice
AU  - Luca Cattani
AU  - Leandro Chiarini
AU  - Alessandra Cipriani
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32687
UR  - https://arxiv.org/abs/2610.32687
AB  - 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.
ER  -