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April 2022 A singular Toeplitz determinant and the discrete tacnode kernel for skew-Aztec rectangles
Mark Adler, Kurt Johansson, Pierre van Moerbeke
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Ann. Appl. Probab. 32(2): 1234-1294 (April 2022). DOI: 10.1214/21-AAP1708

Abstract

Random tilings of geometrical shapes with dominos or lozenges have been a rich source of universal statistical distributions. This paper deals with domino tilings of checker board rectangular shapes such that the top two and bottom two adjacent squares have the same orientation and the two most left and two most right ones as well. It forces these so-called “skew-Aztec rectangles” to have cuts on either side. For large sizes of the domain and upon an appropriate scaling of the location of the cuts, one finds split tacnodes between liquid regions with two distinct adjacent frozen phases descending into the tacnode. Zooming about such split tacnodes, filaments appear between the liquid patches evolving in a bricklike sea of dimers of another type. This work shows that the random fluctuations in a neighborhood of the split tacnode are governed asymptotically by the discrete tacnode kernel, providing strong evidence that this kernel is a universal discrete-continuous limiting kernel occurring naturally whenever we have doubly interlacing patterns. The analysis involves the inversion of a singular Toeplitz matrix which leads to considerable difficulties.

Funding Statement

The first author was supported by Simons Foundation Grants #278931.
The second author was supported by the Swedish Research Council (VR) and grant KAW 2015.0270 of the Knut and Alice Wallenberg Foundation.
The third author was supported by Simons Foundation Grants #280945.

Acknowledgments

We thank Sunil Chhita (Durham University) for the beautiful and especially insightful computer simulations in Figure 6.

Citation

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Mark Adler. Kurt Johansson. Pierre van Moerbeke. "A singular Toeplitz determinant and the discrete tacnode kernel for skew-Aztec rectangles." Ann. Appl. Probab. 32 (2) 1234 - 1294, April 2022. https://doi.org/10.1214/21-AAP1708

Information

Received: 1 December 2019; Revised: 1 March 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

Digital Object Identifier: 10.1214/21-AAP1708

Subjects:
Primary: 35Q53 , 60G60 , 60G65
Secondary: 35Q58 , 60G10

Keywords: Domino tilings , singular Toeplitz determinants , Skew-Aztec rectangles , the discrete tacnode kernel

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.32 • No. 2 • April 2022
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