## Abstract

We present a survey of results on convergence in sandpile models. For a sandpile model on a triangular lattice we prove results similar to the ones known for a square lattice. Namely, consider the sandpile model on the integer points of the plane and put n grains of sand at the origin. Let us begin the process of relaxation: if the number of grains of sand at some vertex z is not less than its valency (in this case we say that the vertex z is unstable), then we move a grain of sand from z to each adjacent vertex, and then repeat this operation as long as there are unstable vertices. We prove that the support of the state (nδ_{0})^{◦} in which the process stabilizes grows at a rate of^{√}n and, after rescaling with coefficient^{√}n, (nδ_{0})^{◦} has a limit in the weak-^{∗} topology. This result was established by Pegden and Smart for the square lattice (where every vertex is connected with four nearest neighbours); we extend it to a triangular lattice (where every vertex is connected with six neighbours).

Original language | English |
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Pages (from-to) | 1651-1673 |

Number of pages | 23 |

Journal | Sbornik Mathematics |

Volume | 214 |

Issue number | 12 |

DOIs | |

State | Published - 2023 |

## Keywords

- discrete Green’s function
- discrete harmonic and superharmonic functions
- sandpile models
- triangular lattice