Cut-off for sandpiles on 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 general method obtaining Green’s function tiling is given, total variation cut-off phenomenon demonstrated under It shown that condition does not affect mixing time for planar tilings. In companion paper, computational methods used to demonstrate an alters D4 lattice in dimension 4, while asymptotic evaluation shows it change cubic Zd all sufficiently large d.