Sandpiles on the square lattice

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.

v3: Minor changes

References in corpus (1)

• Stabilizability and percolation in the infinite volume sandpile model

Cited by in corpus (7)

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• Convergence of the random Abelian sandpile
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• The spectrum of the abelian sandpile model
• The hockey-stick conjecture for activated random walk