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Local geometry of the rough-smooth interface in the two-periodic Aztec diamond

Beffara, Vincent; Chhita, Sunil; Johansson, Kurt

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Authors

Vincent Beffara

Kurt Johansson



Abstract

Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. In a previous paper, the authors found that a certain averaging of height function differences at the rough-smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level lines of the height function. We show, after suitable centering and rescaling, that a point process from these paths converge to the extended Airy kernel point process provided that the natural parameter associated to the two-periodic Aztec diamond is small enough.

Citation

Beffara, V., Chhita, S., & Johansson, K. (2022). Local geometry of the rough-smooth interface in the two-periodic Aztec diamond. Annals of Applied Probability, 32(2), 974-1017. https://doi.org/10.1214/21-aap1701

Journal Article Type Article
Acceptance Date May 26, 2021
Publication Date 2022-04
Deposit Date Jun 1, 2021
Publicly Available Date Jun 20, 2022
Journal Annals of Applied Probability
Print ISSN 1050-5164
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 32
Issue 2
Pages 974-1017
DOI https://doi.org/10.1214/21-aap1701
Publisher URL https:/doi.org/10.1214/21-AAP1701

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Publisher Licence URL
http://creativecommons.org/licenses/by/4.0/

Copyright Statement
This research was funded, in whole or in part, by [UK Engineering and Physical Sciences Research Council, EP/T004290/1]. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant's open access conditions




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