Domino tilings of the Aztec diamond with doubly periodic weightings

Berggren, Tomas

doi:10.1214/20-aop1498

Cited by 11 publications

(17 citation statements)

References 24 publications

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“…These corridors go all the way up to the boundary and the height at the boundary defines the corridor height for all faces in the corridor. 5 Differences between corridor heights on faces gives the (signed) number of paths between faces. For 1 ≤ p ≤ L 2 , 1 ≤ q ≤ L 1 , the intervals {β q } × A p , can be rescaled and put between faces at the rough-smooth boundary, and we can consider the corridor height differences between faces.…”

Section: Informal Statement Of Theoremmentioning

confidence: 99%

“…(B. 5) We have that h a J r p,q,k,1 − h a J r p,q,k,k = i (−1) ε(z i ) I p,q,k (z i ), (B. 6) where i is the sum of all particles in the determinantal point process on L m .…”

Section: Convergence To the Full-plane Smooth Phase The Kasteleyn Mat...mentioning

confidence: 99%

“…Yet another approach based on Wiener-Hopf factorization of matrix-valued symbols was given in [6]. Further developments of these approaches have been particularly fruitful in other models [5,8].…”

Section: Introductionmentioning

confidence: 99%

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

2022

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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.

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Section: Informal Statement Of Theoremmentioning

confidence: 99%

“…(B. 5) We have that h a J r p,q,k,1 − h a J r p,q,k,k = i (−1) ε(z i ) I p,q,k (z i ), (B. 6) where i is the sum of all particles in the determinantal point process on L m .…”

Section: Convergence To the Full-plane Smooth Phase The Kasteleyn Mat...mentioning

confidence: 99%

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

2022

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“…The idea that was introduced in [BD19], and used in [Ber21] and [BD22], is to first find a Wiener-Hopf factorization for B = B + (z)B − (z) (and similarly B = B− (z) B+ (z)), and then continue with the symbol…”

Section: Periodic Weightsmentioning

confidence: 99%

“…The refactorization procedure in this case is equivalent to a Wiener-Hopf factorization of the symbol corresponding to the product of the transition matrices. For various models, such as the two-periodic Aztec diamond and even a class of weightings with higher periodicity [Ber21], the dynamical system determined by the refactorization procedure is periodic and this allows for very explicit double integral formulation for the correlation kernel that can be analyzed asymptotically. In general, tracing this dynamical system is not an easy task.…”

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confidence: 99%

On the domino shuffle and matrix refactorizations

Chhita¹,

Duits²

2022

Preprint

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This paper is motivated by computing correlations for domino tilings of the Aztec diamond. It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. One of the methods, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for twoby-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU-and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. This paper shows that, for arbitrary weightings of the Aztec diamond, the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. In particular, these dynamics can be used to find the inverse of the LGV matrix in the Eynard-Mehta Theorem.

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Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

Assiotis

2020

Ann. Henri Poincaré

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We introduce a space inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari in [7]. We show that for a certain class of initial conditions the point process associated to the dynamics has determinantal correlation functions and we calculate explicitly, in the form of a double contour integral, the correlation kernel for one of the most classical initial conditions, the densely packed. En route to proving this we obtain some results of independent interest on non-intersecting general pure-birth chains, that generalize the Charlier process, the discrete analogue of Dyson's Brownian motion. Finally, these dynamics provide a coupling between the inhomogeneous versions of the TAZRP and PushTASEP particle systems which appear as projections on the left and right edges of the array respectively⋆ ⋆ + 1 ⋆ + 2 ⋆ + 3 ⋆ + 4 ⋆ + 5 ⋆ + 6 Figure 1: A configuration of particles in GT 4 . If the clock of the particle labelled X 3,1 4 rings, which happens at rate λ(⋆ + 1), then the move is blocked since interlacing with X 2,1 4 would be violated. On the other hand, if the clock of the particle X 2,2 4 rings, which happens at rate λ(⋆ + 3), then it jumps to the right by one and instantaneously pushes both X 3,3 4 and X 4,4 4 to the right by one as well, for otherwise the interlacing would break.The right edge particle system is called inhomogeneous PushTASEP, see [7], [6] and also [16] for a q-deformation of the homogeneous case. The fact that this particle system has an underlying determinantal structure is new as well, but is also obtained in the independent work of Petrov [46] that uses different methods.

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Domino tilings of the Aztec diamond with doubly periodic weightings

Cited by 11 publications

References 24 publications

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

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

On the domino shuffle and matrix refactorizations

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

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