Convergence of a sandpile on a triangular lattice under rescaling

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 $(n\delta_0)^\circ$ in which the process stabilizes grows at a rate of $\sqrt n$ and, after rescaling with coefficient $\sqrt n$, $(n\delta_0)^\circ$ 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). Bibliography: 39 titles.