Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 57 min ago
PreprintCC BY 4.0arXiv

Limit shape, avalanches, and stabilisation of abelian sandpiles on comb lattices

  • Robin Kaiser
  • Ecaterina Sava-Huss
  • Julia Überbacher
Open at sourcePDFDOICite

Abstract

We study two aspects of the abelian sandpile model on comb lattices. First, we prove that the infinite-volume limit of the stationary measures is supported on the saturated configuration. We then investigate the shape of avalanches induced by adding a particle at the origin in finite boxes of size (2n+1)×(2n+1) around the origin. In the stationary distribution on these finite boxes, we show that avalanches reach the boundary along the vertical teeth with probability tending to $1$, while the horizontal spread is of order √n. Finally, we establish that the single-source limit shape for abelian sandpiles on the comb lattice agrees with the corresponding limit shapes for the divisible sandpile, internal diffusion-limited aggregation (IDLA), and rotor-router aggregation models. This establishes limit shape universality on the comb lattice.

Related

  1. Avalanches of the Abelian sandpile on the hierarchical lattice
    Luca Cattani, Leandro Chiarini, Alessandra Cipriani
    arXivPreprintarXiv licenceFull textMathematical physicsProbabilityarXiv
  2. Scaling limit of the 3D abelian Yang--Mills Langevin dynamics
    Ilya Chevyrev, Yahui Qu, Hao Shen
    arXivPreprintCC0Full textAnalysis & PDEsMathematical physicsarXiv
  3. Sandpile avalanches in two dimensions are finite
    Ahmed Bou-Rabee
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  4. Active Flux and Polygons: why a stabilisation and how to implement boundary conditions
    Remi Abgrall, Yongle Liu
    arXivPreprintCC BY-NC-ND 4.0Full textNumerical analysis & scientific computingarXiv
  5. Runaway avalanches in plasmas with external electric fields: spatially inhomogeneous case in a perturbation framework
    Ling-Bing He, Richard M. Höfer, Jie Ji +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{kaiser2026limit,
  title = {{Limit shape, avalanches, and stabilisation of abelian sandpiles on comb lattices}},
  author = {Robin Kaiser and Ecaterina Sava-Huss and Julia Überbacher},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.21359},
  eprint = {2609.21359},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21359},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Limit shape, avalanches, and stabilisation of abelian sandpiles on comb lattices
AU  - Robin Kaiser
AU  - Ecaterina Sava-Huss
AU  - Julia Überbacher
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.21359
UR  - https://arxiv.org/abs/2609.21359
AB  - We study two aspects of the abelian sandpile model on comb lattices. First, we prove that the infinite-volume limit of the stationary measures is supported on the saturated configuration. We then investigate the shape of avalanches induced by adding a particle at the origin in finite boxes of size (2n+1)×(2n+1) around the origin. In the stationary distribution on these finite boxes, we show that avalanches reach the boundary along the vertical teeth with probability tending to $1$, while the horizontal spread is of order √n. Finally, we establish that the single-source limit shape for abelian sandpiles on the comb lattice agrees with the corresponding limit shapes for the divisible sandpile, internal diffusion-limited aggregation (IDLA), and rotor-router aggregation models. This establishes limit shape universality on the comb lattice.
ER  -