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April 2022 Local geometry of the rough-smooth interface in the two-periodic Aztec diamond
Vincent Beffara, Sunil Chhita, Kurt Johansson
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Ann. Appl. Probab. 32(2): 974-1017 (April 2022). DOI: 10.1214/21-AAP1701

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.

Funding Statement

SC acknowledges the support of the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/T004290/1. KJ acknowledges the support of the Swedish Research Council (VR) and grant KAW 2015.0270 of the Knut and Alice Wallenberg Foundation.

Acknowledgements

We would like to thank the referees for their careful readings and comments on an earlier version of this paper.

Citation

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Vincent Beffara. Sunil Chhita. Kurt Johansson. "Local geometry of the rough-smooth interface in the two-periodic Aztec diamond." Ann. Appl. Probab. 32 (2) 974 - 1017, April 2022. https://doi.org/10.1214/21-AAP1701

Information

Received: 1 June 2020; Revised: 1 April 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

Digital Object Identifier: 10.1214/21-AAP1701

Subjects:
Primary: 60G55 , 60K35 , 82B20

Keywords: Airy kernel point process , Domino tilings , two-periodic Aztec diamond

Rights: 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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