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

Beffara, Vincent; Chhita, Sunil; Johansson, Kurt

doi:10.1214/21-aap1701

Cited by 7 publications

(3 citation statements)

References 28 publications

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“…(iv) The Airy line ensemble appears as the limit at the boundary between the frozen and liquid regions for a very general class of random lozenge tilings, see [4], also [22,[35][36][37] for earlier convergence results and work on related models. The Airy kernel has also been found at the more delicate liquid-gas boundary in the Aztec diamond [7,8].…”

Section: Introductionmentioning

confidence: 83%

Bulk properties of the Airy line ensemble

Dauvergne

Virág

2021

Ann. Probab.

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The parabolic Airy line ensemble A is a central limit object in the KPZ universality class and related areas. On any compact set K = {1, . . . , k} × [a, a + t], the law of the recentered ensemble A−A(a) has a density X K with respect to the law of k independent Brownian motions. We show thatwhere S is an explicit, tractable, non-negative function of f . We use this formula to show that X K is bounded above by a K-dependent constant, give a sharp estimate on the size of the set where X K < ǫ as ǫ → 0, and prove a large deviation principle for A. We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two-point tail bounds that are stronger than those for Brownian motion. The paper is essentially self-contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.1. Almost surely, L satisfies L i (r) > L i+1 (r) for all pairs (i, r) ∉ 1, k × (0, t).

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Section: Introductionmentioning

confidence: 83%

Bulk properties of the Airy line ensemble

Dauvergne

Virág

2021

Ann. Probab.

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“…For uniformly random domino tilings, we expect that the existing methods can be adapted to show universalities at smooth and cusp points, analogous to [6] and this paper. It would also be interesting to consider random tilings with nonuniform measures, such as the various weighted ones [15,17,24,30,36,55].…”

Section: Proof Ideasmentioning

confidence: 99%

Pearcey universality at cusps of polygonal lozenge tilings

Huang,

Yang,

Zhang

2024

Comm Pure Appl Math

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We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved by Aggarwal‐Huang and Aggarwal‐Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with nonintersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formulas.

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“…Much research focuses on the Airy kernel point processes that have been observed at the boundary [CJ16, Joh17, BCJ18, BCJ22, JM23b]. There has also been progress on finding microscopic boundary paths [BCJ22,JM23a].…”

Section: Introductionmentioning

confidence: 99%

Local correlation functions of the two-periodic weighted Aztec diamond in mesoscopic limit

Bain

2023

Journal of Mathematical Physics

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Here, we study the two-periodic weighted dimer model on the Aztec diamond graph. In the thermodynamic limit when the size of the graph goes to infinity while weights are fixed, the model develops a limit shape with frozen regions near corners, a flat “diamond” in the center with a noncritical (ordered) phase, and a disordered phase separating this diamond and the frozen phase. We show that in the mesoscopic scaling limit, when weights scale in the thermodynamic limit such that the size of the “flat diamond” is of the same order as the correlation length inside the diamond, fluctuations of the height function are described by a new process. We compute asymptotics of the inverse Kasteleyn matrix for vertices in a local neighborhood in this mesoscopic limit.

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

Cited by 7 publications

References 28 publications

Bulk properties of the Airy line ensemble

Bulk properties of the Airy line ensemble

Pearcey universality at cusps of polygonal lozenge tilings

Local correlation functions of the two-periodic weighted Aztec diamond in mesoscopic limit

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