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

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.

Dates and versions

hal-02572271 , version 1 (13-05-2020)

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Vincent Beffara, Sunil Chhita, Kurt Johansson. Local Geometry of the rough-smooth interface in the two-periodic Aztec diamond. 2020. ⟨hal-02572271⟩
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