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.

arXiv commentv3: Minor changes

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Cited by in corpus (7)

• Cut-off for sandpiles on tiling graphs
• Convergence of the random Abelian sandpile
• Exact sampling and fast mixing of Activated Random Walk
• Nonstationary but quasisteady states in Self-organized Criticality
• How far do Activated Random Walkers spread from a single source?
• The spectrum of the abelian sandpile model
• The hockey-stick conjecture for activated random walk