Double interlacing in random tiling models
Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid–liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain boundary, for large-sized domains, the...
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craippubl:10.1063/5.0093542 2024-02-11T10:00:48+01:00 Double interlacing in random tiling models Adler, Mark van Moerbeke, Pierre Simons Foundation NSF 2023 http://dx.doi.org/10.1063/5.0093542 https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/5.0093542/16794499/033509_1_online.pdf en eng AIP Publishing Journal of Mathematical Physics volume 64, issue 3 ISSN 0022-2488 1089-7658 Mathematical Physics Statistical and Nonlinear Physics journal-article 2023 craippubl https://doi.org/10.1063/5.0093542 2024-01-26T09:46:42Z Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid–liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain boundary, for large-sized domains, the tiles of a certain shape form a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a Gaussian unitary ensemble-matrix. Introducing non-convexities in large domains may lead to the appearance of several interacting liquid regions: They can merely touch, leading to either a split tacnode (hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the non-convex domains and probing about such split tacnodes, filaments, evolving in a bricklike sea of dimers of another type, will connect the liquid patches. Nearby, the tiling fluctuations are governed by a discrete tacnode kernel—i.e., a determinantal point process on a doubly interlacing set of dots belonging to a discrete array of parallel lines. This kernel enables us to compute the joint distribution of the dots along those lines. This kernel appears in two very different models: (i) domino tilings of skew-Aztec rectangles and (ii) lozenge tilings of hexagons with cuts along opposite edges. Soft tacnodes appear when two arctic curves gently touch each other amid a bricklike sea of dimers of one type, unlike the split tacnode. We hope that this largely expository paper will provide a view on the subject and be accessible to a wider audience. Article in Journal/Newspaper Arctic AIP Publishing Arctic Journal of Mathematical Physics 64 3 033509 |
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English |
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Mathematical Physics Statistical and Nonlinear Physics |
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Mathematical Physics Statistical and Nonlinear Physics Adler, Mark van Moerbeke, Pierre Double interlacing in random tiling models |
topic_facet |
Mathematical Physics Statistical and Nonlinear Physics |
description |
Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid–liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain boundary, for large-sized domains, the tiles of a certain shape form a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a Gaussian unitary ensemble-matrix. Introducing non-convexities in large domains may lead to the appearance of several interacting liquid regions: They can merely touch, leading to either a split tacnode (hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the non-convex domains and probing about such split tacnodes, filaments, evolving in a bricklike sea of dimers of another type, will connect the liquid patches. Nearby, the tiling fluctuations are governed by a discrete tacnode kernel—i.e., a determinantal point process on a doubly interlacing set of dots belonging to a discrete array of parallel lines. This kernel enables us to compute the joint distribution of the dots along those lines. This kernel appears in two very different models: (i) domino tilings of skew-Aztec rectangles and (ii) lozenge tilings of hexagons with cuts along opposite edges. Soft tacnodes appear when two arctic curves gently touch each other amid a bricklike sea of dimers of one type, unlike the split tacnode. We hope that this largely expository paper will provide a view on the subject and be accessible to a wider audience. |
author2 |
Simons Foundation NSF |
format |
Article in Journal/Newspaper |
author |
Adler, Mark van Moerbeke, Pierre |
author_facet |
Adler, Mark van Moerbeke, Pierre |
author_sort |
Adler, Mark |
title |
Double interlacing in random tiling models |
title_short |
Double interlacing in random tiling models |
title_full |
Double interlacing in random tiling models |
title_fullStr |
Double interlacing in random tiling models |
title_full_unstemmed |
Double interlacing in random tiling models |
title_sort |
double interlacing in random tiling models |
publisher |
AIP Publishing |
publishDate |
2023 |
url |
http://dx.doi.org/10.1063/5.0093542 https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/5.0093542/16794499/033509_1_online.pdf |
geographic |
Arctic |
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Arctic |
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Arctic |
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Arctic |
op_source |
Journal of Mathematical Physics volume 64, issue 3 ISSN 0022-2488 1089-7658 |
op_doi |
https://doi.org/10.1063/5.0093542 |
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Journal of Mathematical Physics |
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64 |
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3 |
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033509 |
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1790596514016395264 |