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The spectrum of the abelian sandpile model


Authors: Robert Hough and Hyojeong Son
Journal: Math. Comp. 90 (2021), 441-469
MSC (2010): Primary 82C20, 60B15, 60J10
DOI: https://doi.org/10.1090/mcom/3565
Published electronically: August 26, 2020
MathSciNet review: 4166468
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Abstract: In their previous work, the authors studied the abelian sandpile model on graphs constructed from a growing piece of a plane or space tiling, given periodic or open boundary conditions, and identified spectral parameters which govern the asymptotic spectral gap and asymptotic mixing time. This paper gives a general method of determining the spectral parameters either computationally or asymptotically, and determines the spectral parameters in specific examples.


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Additional Information

Robert Hough
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York, 11794
MR Author ID: 873503
Email: robert.hough@stonybrook.edu

Hyojeong Son
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York, 11794
Address at time of publication: Department of Mathematics and Statistics, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130-4899
ORCID: 0000-0001-5125-3364
Email: hyojeong.son@wustl.edu

DOI: https://doi.org/10.1090/mcom/3565
Keywords: Abelian sandpile model, random walk on a group, spectral gap, cut-off phenomenon
Received by editor(s): May 20, 2019
Received by editor(s) in revised form: February 3, 2020, and May 13, 2020
Published electronically: August 26, 2020
Additional Notes: 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.
The second author was supported by a fellowship from the Summer Math Foundation at Stony Brook.
Article copyright: © Copyright 2020 American Mathematical Society