Sandpile dynamics on periodic tiling graphs

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 method of computing the Green's function and spectral gap is given for various tilings and a cut-off phenomenon in the mixing is demonstrated under general conditions. It is shown that the boundary condition does not affect the mixing time for planar tilings, but that the mixing time is altered for the $\mathrm{D4}$ lattice in dimension 4. For all sufficiently large $d$, the boundary condition does not affect the mixing of the cubic lattice $\mathbb{Z}^d$.