Tropical curves in sandpile models
- 1Guangdong Technion Israel Institute of Technology (GTIIT), 241 Daxue Road, Shantou, Guangdong Province 515603, PR China
- 2Technion-Israel Institute of Technology, Haifa, 32000, Haifa District, Israel
- 3Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria
Abstract
A sandpile is a cellular automaton on a graph that evolves by the following toppling rule: if the number of grains at a vertex is at least its valency, then this vertex sends one grain to each of its neighbors. In the study of pattern formation in sandpiles on large subgraphs of the standard square lattice, S. Caracciolo, G. Paoletti, and A. Sportiello experimentally observed that the result of the relaxation of a small perturbation of the maximal stable state contains a clearly visible thin balanced graph formed by its deviation (less than maximum) set. Such graphs are known as tropical curves. During the early stage of our research, we noticed that these tropical curves are approximately scale-invariant, that is the deviation set mimics an extremal tropical curve depending on the domain on the plane and the positions of the perturbation points, but not on the mesh of the lattice. In this paper, we rigorously formulate these two facts in the form of a scaling limit theorem and prove it. We rely on the theory of tropical analytic series, which is used to describe the global features of the sandpile dynamics, and on the theory of smoothings of discrete superharmonic functions, which handles local questions.
Related
BibTeX
@article{kalinin2026tropical,
title = {{Tropical curves in sandpile models}},
author = {Nikita Sergeevich Kalinin and Mikhail Shkolnikov},
journal = {Advances in Mathematics},
year = {2026},
month = jul,
doi = {10.1016/j.aim.2026.111127},
url = {https://doi.org/10.1016/j.aim.2026.111127},
note = {CC BY},
}
RIS
TY - JOUR
TI - Tropical curves in sandpile models
AU - Nikita Sergeevich Kalinin
AU - Mikhail Shkolnikov
PY - 2026
DA - 2026/07/08
JO - Advances in Mathematics
DO - 10.1016/j.aim.2026.111127
UR - https://doi.org/10.1016/j.aim.2026.111127
AB - A sandpile is a cellular automaton on a graph that evolves by the following toppling rule: if the number of grains at a vertex is at least its valency, then this vertex sends one grain to each of its neighbors. In the study of pattern formation in sandpiles on large subgraphs of the standard square lattice, S. Caracciolo, G. Paoletti, and A. Sportiello experimentally observed that the result of the relaxation of a small perturbation of the maximal stable state contains a clearly visible thin balanced graph formed by its deviation (less than maximum) set. Such graphs are known as tropical curves. During the early stage of our research, we noticed that these tropical curves are approximately scale-invariant, that is the deviation set mimics an extremal tropical curve depending on the domain on the plane and the positions of the perturbation points, but not on the mesh of the lattice. In this paper, we rigorously formulate these two facts in the form of a scaling limit theorem and prove it. We rely on the theory of tropical analytic series, which is used to describe the global features of the sandpile dynamics, and on the theory of smoothings of discrete superharmonic functions, which handles local questions.
SN - 0001-8708
SN - 1090-2082
ER -