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

Beffara, Vincent; Chhita, Sunil; Johansson, Kurt

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
• Public URL: https://durham-repository.worktribe.com/output/1241796
• Publisher URL: https:/doi.org/10.1214/21-AAP1701