Double Aztec Diamonds and the Tacnode Process
Discrete and continuous non-intersecting random processes have given rise to critical "infinite dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordere...
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ftdatacite:10.48550/arxiv.1112.5532 2023-05-15T14:58:41+02:00 Double Aztec Diamonds and the Tacnode Process Adler, Mark Johansson, Kurt van Moerbeke, Pierre 2011 https://dx.doi.org/10.48550/arxiv.1112.5532 https://arxiv.org/abs/1112.5532 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Probability math.PR Mathematical Physics math-ph Combinatorics math.CO FOS Mathematics FOS Physical sciences Preprint Article article CreativeWork 2011 ftdatacite https://doi.org/10.48550/arxiv.1112.5532 2022-04-01T14:00:42Z Discrete and continuous non-intersecting random processes have given rise to critical "infinite dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the "tacnode process". It is also shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily. : 80 pages, 18 figures Report Arctic DataCite Metadata Store (German National Library of Science and Technology) Arctic |
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Probability math.PR Mathematical Physics math-ph Combinatorics math.CO FOS Mathematics FOS Physical sciences |
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Probability math.PR Mathematical Physics math-ph Combinatorics math.CO FOS Mathematics FOS Physical sciences Adler, Mark Johansson, Kurt van Moerbeke, Pierre Double Aztec Diamonds and the Tacnode Process |
topic_facet |
Probability math.PR Mathematical Physics math-ph Combinatorics math.CO FOS Mathematics FOS Physical sciences |
description |
Discrete and continuous non-intersecting random processes have given rise to critical "infinite dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the "tacnode process". It is also shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily. : 80 pages, 18 figures |
format |
Report |
author |
Adler, Mark Johansson, Kurt van Moerbeke, Pierre |
author_facet |
Adler, Mark Johansson, Kurt van Moerbeke, Pierre |
author_sort |
Adler, Mark |
title |
Double Aztec Diamonds and the Tacnode Process |
title_short |
Double Aztec Diamonds and the Tacnode Process |
title_full |
Double Aztec Diamonds and the Tacnode Process |
title_fullStr |
Double Aztec Diamonds and the Tacnode Process |
title_full_unstemmed |
Double Aztec Diamonds and the Tacnode Process |
title_sort |
double aztec diamonds and the tacnode process |
publisher |
arXiv |
publishDate |
2011 |
url |
https://dx.doi.org/10.48550/arxiv.1112.5532 https://arxiv.org/abs/1112.5532 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1112.5532 |
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1766330814347345920 |