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Sandpiles on the Square Lattice

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    Robert D. Hough

  2. Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY, 14853, USA

    Daniel C. Jerison & Lionel Levine

  3. Department of Mathematics, Tel Aviv University, Tel Aviv, Israel

    Daniel C. Jerison

Authors
  1. Robert D. Hough
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  2. Daniel C. Jerison
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  3. Lionel Levine
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Correspondence to Daniel C. Jerison.

Additional information

Communicated by H. Duminil-Copin

This material is based upon work supported by the National Science Foundation under Agreements No. DMS-1128155, http://www.nsf.gov/awardsearch/showAward?AWD_ID=1455272DMS-1455272, DMS-1712682, and DMS-1802336. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Hough, R.D., Jerison, D.C. & Levine, L. Sandpiles on the Square Lattice. Commun. Math. Phys. 367, 33–87 (2019). https://doi.org/10.1007/s00220-019-03408-5

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