Directed nonabelian sandpile models on trees

TL;DR

This paper introduces two new nonabelian sandpile models on directed trees, analyzing their stationary distributions and convergence properties using advanced algebraic methods.

Contribution

It defines two novel nonabelian sandpile models on directed trees and provides explicit formulas for their stationary distributions and spectral properties.

Findings

Stationary distribution for Trickle-down model is of product form.

Eigenvalues and convergence rates are derived for Landslide model.

Models extend understanding of nonabelian sandpile dynamics.

Abstract

We define two general classes of nonabelian sandpile models on directed trees (or arborescences) as models of nonequilibrium statistical phenomena. These models have the property that sand grains can enter only through specified reservoirs, unlike the well-known abelian sandpile model. In the Trickle-down sandpile model, sand grains are allowed to move one at a time. For this model, we show that the stationary distribution is of product form. In the Landslide sandpile model, all the grains at a vertex topple at once, and here we prove formulas for all eigenvalues, their multiplicities, and the rate of convergence to stationarity. The proofs use wreath products and the representation theory of monoids.