Sandpiles on the Square Lattice

  • Robert D. Hough
  • Daniel C. JerisonEmail author
  • Lionel Levine


We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice \({\mathbb{Z}^2}\) . We also determine the asymptotic spectral gap, asymptotic mixing time, and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus \({\left(\mathbb{Z}/m\mathbb{Z}\right)^2}\) . The techniques use analysis of the space of functions on \({\mathbb{Z}^2}\) which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in \({\ell^p(\mathbb{Z}^2)}\) as linear combinations of certain discrete derivatives of Green’s functions, extending a result of Schmidt and Verbitskiy (Commun Math Phys 292(3):721–759, 2009. arXiv:0901.3124 [math.DS]).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Robert D. Hough
    • 1
  • Daniel C. Jerison
    • 2
    • 3
    Email author
  • Lionel Levine
    • 2
  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA
  3. 3.Department of MathematicsTel Aviv UniversityTel AvivIsrael

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