# Cut-off for sandpiles on tiling graphs

## TL;DR

This paper investigates the mixing times of sandpile dynamics on tiling graphs, establishing a general Green's function method and demonstrating a total variation cut-off phenomenon under broad conditions.

## Contribution

It introduces a general approach to compute Green's functions on tiling graphs and proves a total variation cut-off phenomenon for sandpile dynamics under various boundary conditions.

## Key findings

- Boundary conditions do not affect mixing time on planar tilings.
- Asymptotic mixing time remains unchanged for large dimensions in cubic lattices.
- Computational evidence shows altered mixing times for the D4 lattice in dimension 4.

## Abstract

Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling either torus or open boundary conditions. A general method of obtaining the Green's function of the tiling is given, and a total variation cut-off phenomenon is demonstrated under general conditions. It is shown that the boundary condition does not affect the mixing time for planar tilings, nor does it change the asymptotic mixing time for the cubic lattice $\zed^d$ for all sufficiently large $d$. In a companion paper, computational methods are used to demonstrate that the mixing time is altered for the $\Dfour$ lattice in dimension 4.