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We introduce a new class of nonequilibrium statistical mechanical models on rooted trees related to the well-known abelian sandpile model. Each vertex of the tree has a certain threshold, which represents the number of grains of sand it can hold. Sand enters the tree from the leaves, topples along the path to the root, and exits at the root.

We will define two kinds of models within this framework. In the Trickle-down model, sand grains topple one at a time. We will prove that the stationary distribution in this case is exactly a product measure. In the Landslide model, all the sand at a given vertex topples at once. We will give explicit formulas for all the eigenvalues and their multiplicities of the transition matrix.

This is joint work with Nicolas Thiery, Anne Schilling and Ben Steinberg.