Abstract
We introduce a natural stochastic extension, called SSP, of the abelian sandpile model (ASM), which shares many mathematical properties with ASM, yet radically differs in its physical behavior, for example in terms of the shape of the steady state and of the avalanche size distribution. We establish a basic theory of SSP analogous to that of ASM, and present a brief numerical study of its behavior. Our original motivation for studying SSP stems from its connection to the LLL algorithm established in another work by Ding et al. (LLL and stochastic sandpile models, https://sites.google.com/view/seungki/). The importance of understanding how LLL works cannot be stressed more, especially from the point of view of lattice-based cryptography. We believe SSP serves as a tractable toy model of LLL that would help further our understanding of it.
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Notes
The only reason to impose the condition \((J, I) = 1\) is to prevent the pile heights at each site from being concentrated on a select few congruence classes modulo I; it is not so much an essential condition as a cosmetic one.
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Acknowledgements
We thank Deepak Dhar, Phong Nguyen, and Su-Chan Park for helpful comments and suggestions.
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Communicated by Eric A. Carlen.
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Kim, S., Wang, Y. A Stochastic Variant of the Abelian Sandpile Model. J Stat Phys 178, 711–724 (2020). https://doi.org/10.1007/s10955-019-02453-7
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DOI: https://doi.org/10.1007/s10955-019-02453-7