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Asymptotic geometry of discrete interlaced patterns: Part I
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2015 (English)In: International Journal of Mathematics, ISSN 0129-167XArticle in journal (Refereed) Published
Abstract [en]

A discrete Gelfand–Tsetlin pattern is a configuration of particles in ℤ2. The particles are arranged in a finite number of consecutive rows, numbered from the bottom. There is one particle on the first row, two particles on the second row, three particles on the third row, etc., and particles on adjacent rows satisfy an interlacing constraint. We consider the uniform probability measure on the set of all discrete Gelfand–Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform probability measure on the set of all tilings of certain polygons with lozenges. We prove a determinantal structure, and calculate the correlation kernel. We consider the asymptotic behavior of the system as the size increases under the assumption that the empirical distribution of the deterministic particles on the top row converges weakly. We consider the asymptotic "shape" of such systems. We provide parameterizations of the asymptotic boundaries and investigate the local geometric properties of the resulting curves. We show that the boundary can be partitioned into natural sections which are determined by the behavior of the roots of a function related to the correlation kernel. This paper should be regarded as a companion piece to the paper [E. Duse and A. Metcalfe, Asymptotic geometry of discrete interlaced patterns: Part II, in preparation], in which we resolve some of the remaining issues. Both of these papers serve as background material for the papers [E. Duse and A. Metcalfe, Universal edge fluctuations of discrete interlaced particle systems, in preparation; E. Duse and K. Johansson and A. Metcalfe, Cusp Airy process of discrete interlaced particle systems, in preparation], in which we examine the edge asymptotic behavior.

Place, publisher, year, edition, pages
Singapore: World Scientific, 2015.
Keyword [en]
Random tilings, random matrices, determinantal point processing, universality
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-176651DOI: 10.1142/S0129167X15500937ISI: 000363419500007Scopus ID: 2-s2.0-84945462632OAI: oai:DiVA.org:kth-176651DiVA: diva2:868193
Funder
Knut and Alice Wallenberg Foundation, 2010.0063
Note

QC 20151110

Available from: 2015-11-09 Created: 2015-11-09 Last updated: 2017-12-01Bibliographically approved
In thesis
1. On Uniformly Random Discrete Interlacing Systems
Open this publication in new window or tab >>On Uniformly Random Discrete Interlacing Systems
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns uniformly random discrete interlacing particle sys-tems and their connections to certain random lozenge tiling models. In par-ticular it contains the first derivation of a relatively unknown universal scalinglimit, which we call the Cusp-Airy process, of certain lozenge tiling modelsat a cusp point. In addition it contains a characterization of the geometryof the macroscopic behavior of uniformly random discrete interlaced parti-cle systems that, although not complete, shows many new and interestingfeatures.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. vii, 56 p.
Series
TRITA-MAT-A, 2015:12
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-176652 (URN)978-91-7595-732-6 (ISBN)
Public defence
2015-12-04, Sal D3, Lindstedtsvägen 5, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20151110

Available from: 2015-11-10 Created: 2015-11-09 Last updated: 2015-11-10Bibliographically approved

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