Convergence of a sandpile on a triangular lattice under rescaling

Arkadiy A. Aliev, Nikita S. Kalinin

Research output: Contribution to journalArticlepeer-review

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 (nδ0) in which the process stabilizes grows at a rate ofn and, after rescaling with coefficientn, (nδ0) 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 languageEnglish
Pages (from-to)1651-1673
Number of pages23
JournalSbornik Mathematics
Volume214
Issue number12
DOIs
StatePublished - 2023

Keywords

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

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