Convergence of a sandpile model on a triangular lattice

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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 of √n and, after rescaling with coefficient √n, (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).

About the authors

Arkadiy Artemovich Aliev

Saint-Petersburg State University, Department of Mathematics and Computer Science

Author for correspondence.
Email: nikaanspb@gmail.com

Nikita Sergeevich Kalinin

Guangdong Technion Israel Institute of Technology

Email: nikaanspb@gmail.com

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