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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ftunistlouisbrus:oai:dial.uclouvain.be:boreal:159553 2024-05-12T07:59:34+00:00 Double Aztec diamonds and the tacnode process Adler, Mark Van Moerbeke, Pierre Johansson, Kurt UCL - SST/ICTM - Institute of Information and Communication Technologies, Electronics and Applied Mathematics 2014 http://hdl.handle.net/2078.1/159553 https://doi.org/10.1016/j.aim.2013.10.012 unknown Elsevier boreal:159553 http://hdl.handle.net/2078.1/159553 doi:10.1016/j.aim.2013.10.012 urn:EISSN:1090-2082 urn:ISSN:0001-8708 info:eu-repo/semantics/restrictedAccess Advances in mathematics, Vol. 252, p. 518-571 (2014) QA1 info:eu-repo/semantics/article 2014 ftunistlouisbrus https://doi.org/10.1016/j.aim.2013.10.012 2024-04-18T17:50:13Z 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. © 2013 Elsevier Inc. Article in Journal/Newspaper Arctic DIAL@USL-B (Université Saint-Louis, Bruxelles) Arctic Advances in Mathematics 252 518 571 |
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QA1 Adler, Mark Van Moerbeke, Pierre Johansson, Kurt Double Aztec diamonds and the tacnode process |
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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. © 2013 Elsevier Inc. |
author2 |
UCL - SST/ICTM - Institute of Information and Communication Technologies, Electronics and Applied Mathematics |
format |
Article in Journal/Newspaper |
author |
Adler, Mark Van Moerbeke, Pierre Johansson, Kurt |
author_facet |
Adler, Mark Van Moerbeke, Pierre Johansson, Kurt |
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 |
Elsevier |
publishDate |
2014 |
url |
http://hdl.handle.net/2078.1/159553 https://doi.org/10.1016/j.aim.2013.10.012 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_source |
Advances in mathematics, Vol. 252, p. 518-571 (2014) |
op_relation |
boreal:159553 http://hdl.handle.net/2078.1/159553 doi:10.1016/j.aim.2013.10.012 urn:EISSN:1090-2082 urn:ISSN:0001-8708 |
op_rights |
info:eu-repo/semantics/restrictedAccess |
op_doi |
https://doi.org/10.1016/j.aim.2013.10.012 |
container_title |
Advances in Mathematics |
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252 |
container_start_page |
518 |
op_container_end_page |
571 |
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1798840939640258560 |