Abelian sandpile model

Example

Let $G$ be a finite connected graph without loops, with a distinguished vertex $s$, the sink. Let $V$ be the set of the other vertices. For $x \in V$, let $\deg(x)$ be its degree in $G$. A sandpile is a map $\eta \colon V \to \Z_{\ge 0}$, the number of grains at each vertex. A vertex $x$ is unstable if $\eta(x) \ge \deg(x)$. Toppling an unstable vertex $x$ removes $\deg(x)$ grains from $x$ and adds one grain to each neighbour of $x$ for each edge joining them. Grains sent to $s$ are lost. A sandpile is stable if no vertex is unstable. The abelian sandpile model is a Markov chain on stable sandpiles. At each step, one grain is added at a uniformly random vertex of $V$, and unstable vertices are toppled until the sandpile is stable.

Check on the triangle with vertices $1, 2, s$, where every vertex has degree $2$. Start from the stable sandpile $(\eta(1), \eta(2)) = (1, 1)$ and add a grain at $1$, which gives $(2, 1)$. Vertex $1$ topples: $(0, 2)$, and one grain goes to $s$. Vertex $2$ topples: $(1, 0)$, and one grain goes to $s$. The result $(1, 0)$ is stable, and the process stops.