Communications in Mathematical Physics

, Volume 335, Issue 3, pp 1065–1098 | Cite as

Directed Nonabelian Sandpile Models on Trees

  • Arvind Ayyer
  • Anne Schilling
  • Benjamin Steinberg
  • Nicolas M. Thiéry
Article

Abstract

We define two general classes of nonabelian sandpile models on directed trees (or arborescences), as models of nonequilibrium statistical physics. Unlike usual applications of the well-known abelian sandpile model, these models have the property that sand grains can enter only through specified reservoirs.

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Arvind Ayyer
    • 1
    • 2
  • Anne Schilling
    • 2
  • Benjamin Steinberg
    • 3
  • Nicolas M. Thiéry
    • 4
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Department of MathematicsUC DavisDavisUSA
  3. 3.Department of MathematicsCity College of New YorkNew YorkUSA
  4. 4.Laboratoire de Recherche en Informatique - CNRS: UMR 8623Université Paris-SudOrsayFrance

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