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. In a companion paper, computational methods are used to demonstrate that an open boundary condition alters the mixing time for the D4 lattice in dimension 4, while an asymptotic evaluation shows that it does not change the asymptotic mixing time for the cubic lattice for all sufficiently large d.
Acknowledgments
This material is based upon work supported by the National Science Foundation under agreements No. DMS-1712682 and DMS-1802336. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Hyojeong Son was supported by a fellowship from the Summer Math Foundation at Stony Brook.
Citation
Robert Hough. Hyojeong Son. "Cut-off for sandpiles on tiling graphs." Ann. Probab. 49 (2) 671 - 731, March 2021. https://doi.org/10.1214/20-AOP1458
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