Sandpiles on the Square Lattice

We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice 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 (ℤ/mℤ)^2 . The techniques use analysis of the space of functions on ℤ^2 which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in ℓ^p(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]).