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Double Aztec diamonds and the tacnode process
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-2943-7006
2014 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 252, 518-571 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2014. Vol. 252, 518-571 p.
Keyword [en]
Domino tilings, Aztec diamonds, Dyson's Brownian motion, Airy and tacnode processes, Extended kernels
National Category
URN: urn:nbn:se:kth:diva-141956DOI: 10.1016/j.aim.2013.10.012ISI: 000330153100019ScopusID: 2-s2.0-84889675905OAI: diva2:699632
Swedish Research CouncilKnut and Alice Wallenberg Foundation, KAW 2010.0063

QC 20140228

Available from: 2014-02-28 Created: 2014-02-27 Last updated: 2014-02-28Bibliographically approved

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