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On Uniformly Random Discrete Interlacing Systems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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: urn:nbn:se:kth:diva-176652ISBN: 978-91-7595-732-6 (print)OAI: oai:DiVA.org:kth-176652DiVA: diva2:868196
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
List of papers
1. Asymptotic geometry of discrete interlaced patterns: Part I
Open this publication in new window or tab >>Asymptotic geometry of discrete interlaced patterns: Part I
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
Random tilings, random matrices, determinantal point processing, universality
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-176651 (URN)10.1142/S0129167X15500937 (DOI)000363419500007 ()2-s2.0-84945462632 (Scopus ID)
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
2. Asymptotic geometry of discrete interlaced patterns: Part II
Open this publication in new window or tab >>Asymptotic geometry of discrete interlaced patterns: Part II
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study the boundary of the liquid region L in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line (x,1), x∈R≡∂H. We assume that the initial particle configuration converges weakly to a limiting density ϕ(x), 0≤ϕ≤1. The liquid region is given by a homeomorphism WL:L→H, the upper half plane, and we consider the extension of W−1L to H¯¯¯. Part of ∂L is given by a curve, the edge E, parametrized by intervals in ∂H, and this corresponds to points where ϕ is identical to 0 or 1. If 0<ϕ<1, the non-trivial support, there are two cases. Either W−1L(w) has the limit (x,1) as w→x non-tangentially and we have a \emph{regular point}, or we have what we call a singular point. In this case W−1L does not extend continuously to H¯¯¯. Singular points give rise to parts of ∂L not given by E and which can border a frozen region, or be "inside" the liquid region. This shows that in general the boundary of ∂L can be very complicated. We expect that on the singular parts of ∂L we do not get a universal point process like the Airy or the extended Sine kernel point processes. Furthermore, E and the singular parts of ∂L are shocks of the complex Burgers equation.

National Category
Natural Sciences
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-176649 (URN)
Funder
Knut and Alice Wallenberg Foundation, 2010.0063
Note

QC 20151110

Available from: 2015-11-09 Created: 2015-11-09 Last updated: 2015-11-10Bibliographically approved
3. The Cusp-Airy Process
Open this publication in new window or tab >>The Cusp-Airy Process
(English)Manuscript (preprint) (Other academic)
Abstract [en]

At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point.

National Category
Natural Sciences
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-176650 (URN)
Funder
Knut and Alice Wallenberg Foundation, 2010.0063
Note

QC 20151110

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

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