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The domino shuffling algorithm and Anisotropic KPZ stochastic growth

Chhita, Sunil; Toninelli, Fabio Lucio

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Authors

Fabio Lucio Toninelli



Abstract

The domino-shuffling algorithm [EKLP92a, EKLP92b, Pro03] can be seen as a stochastic process describing the irreversible growth of a (2+1)-dimensional discrete interface [CT19, Zha18]. Its stationary speed of growth ๐‘ฃ๐š (๐œŒ) depends on the average interface slope ๐œŒ, as well as on the edge weights ๐š , that are assumed to be periodic in space. We show that this growth model belongs to the Anisotropic KPZ class [Ton18, Wol91]: one has det[๐ท2๐‘ฃ๐š (๐œŒ)]<0 and the height fluctuations grow at most logarithmically in time. Moreover, we prove that ๐ท๐‘ฃ๐š (๐œŒ) is discontinuous at each of the (finitely many) smooth (or โ€œgaseousโ€) slopes ๐œŒ; at these slopes, fluctuations do not diverge as time grows. For a special case of spatially 2โˆ’periodic weights, analogous results have been recently proven [CT19] via an explicit computation of ๐‘ฃ๐š (๐œŒ). In the general case, such a computation is out of reach; instead, our proof goes through a relation between the speed of growth and the limit shape of domino tilings of the Aztec diamond.

Citation

Chhita, S., & Toninelli, F. L. (2021). The domino shuffling algorithm and Anisotropic KPZ stochastic growth. Annales Henri Lebesgue, 4, 1005-1034. https://doi.org/10.5802/ahl.95

Journal Article Type Article
Acceptance Date Oct 23, 2020
Publication Date 2021
Deposit Date Oct 30, 2020
Publicly Available Date Oct 15, 2021
Journal Annales Henri Lebesgue
Publisher ร‰cole Normale Supรฉrieure de Rennes
Peer Reviewed Peer Reviewed
Volume 4
Pages 1005-1034
DOI https://doi.org/10.5802/ahl.95

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