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

Beffara, Vincent and Chhita, Sunil and Johansson, Kurt (2022) 'Local geometry of the rough-smooth interface in the two-periodic Aztec diamond.', Annals of Applied Probability, 32 (2). pp. 974-1017.


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.

Item Type:Article
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Publisher 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
Date accepted:26 May 2021
Date deposited:01 June 2021
Date of first online publication:2021
Date first made open access:20 June 2022

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