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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Bibliographic Details
Published in:Advances in Mathematics
Main Authors: Adler, Mark, Van Moerbeke, Pierre, Johansson, Kurt
Other Authors: UCL - SST/ICTM - Institute of Information and Communication Technologies, Electronics and Applied Mathematics
Format: Article in Journal/Newspaper
Language:unknown
Published: Elsevier 2014
Subjects:
QA1
Online Access:http://hdl.handle.net/2078.1/159553
https://doi.org/10.1016/j.aim.2013.10.012
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Summary: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.