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On determinantal point processes and random tilings with doubly periodic weights
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is dedicated to asymptotic analysis of determinantal point processes originating from random matrix theory and random tiling models. Our main interest lies in random tilings of planar domains with doubly periodic weights.

Uniformly distributed random tiling models are known to be a very rich class of models where many interesting phenomena can be observed. These models have therefore been under investigation for many years and many aspects of the models are by now well understood. Random tiling models with doubly periodic weights are in fact an even richer class of models. However, these models are much more difficult to analyze and for a thorough study of their behavior new ideas are needed. This thesis increases the understanding of random tiling models with doubly periodic weights.

The thesis consists of three papers and two chapters; one introductory and background chapter and one chapter giving an overview of the papers.

Paper A deals with linear statistics of the thinned Circular Unitary Ensemble and the thinned sine process. The thinning creates a transition from the Circular Unitary Ensemble respectively sine process to the Poisson process. We study a part of these transitions in detail.

In Papers B and C we study random tiling models with doubly periodic weights. These two papers constitute the main contribution of this thesis.

In Paper B we give a general method how to analyze a large family of random tiling models. In particular, we provide a double integral formula for the correlation kernel in terms of a Wiener-Hopf factorization of an associated matrix-valued function. We also present a recursive method on how to construct the Wiener-Hopf factorization.

The method developed in Paper B is used in Paper C to analyze the 2×k-periodic Aztec diamond. More precisely, we derive the correlation kernel for the Aztec diamond of finite size and give a detailed description of the model as the size tends to infinity.

Abstract [sv]

Denna avhandling är ägnad åt asymptotisk analys av determinantprocesserhärstammandes från slumpmatristeori och slumpmässiga plattläggningsmodeller.Vårt huvudintresse är slumpmässiga plattläggningar av platta områdenmed dubbelt periodiska vikter.Likformigt fördelade plattläggningsmodeller är kända för att vara en mycketrik klass av modeller där många intressanta fenomen kan observeras. Dessamodeller har därför utforskats under många år och numera är många aspekterav modellerna välutredda. Slumpmässiga plattläggningsmodeller med dubbeltperiodiska vikter är en till och med rikare klass av modeller. Dessa modeller ärdock mycket svårare att analysera och för en genomgående studie av deras beteendekrävs nya idéer. Denna avhandling ökar förståelsen för slumpmässigaplattläggningsmodeller med dubbelt periodiska vikter.Denna avhandling består av tre artiklar och två kapitel; ett introduktionsochbakgrundskapitel och ett kapitel som ger en översikt av artiklarna.Artikel A analyserar lineära statistikor av en förtunning av den cirkuläraunitära ensemblen och en förtunning av sinusprocessen. Förtunningen skaparen övergång från den cirkulära unitära ensemblen respektive sinusprocessentill Poissonprocessen. Vi analyserar en del av denna övergång i detalj.I Artiklarna B och C studeras slumpmässiga plattläggningsmodeller meddubbelt periodiska vikter. Dessa två artiklar utgör huvudbidraget av dennaavhandling.I Artikel B utvecklar vi en generell metod för att analysera en stor familjav slumpmässiga plattläggningsmodeller. I synnerhet bestämmer vi en formelför korrelationskärnan i termer av en dubbelintegral och en Wiener–Hopffaktorisering av en tillhörande matrisvärd funktion. Vi ger också en rekursivmetod för att konstruera Wiener–Hopf faktoriseringen.I Artikel C används metoden som utvecklats i Artikel B för att analyseraden 2 × k-periodiska aztetiska diamanten. Mer exakt så härleder vi korrelationskärnanför den aztetiska diamanten av ändlig storlek och ger en detaljeradbeskrivning av modellen då storleken går mot oändligheten.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2020. , p. 50
Series
TRITA-SCI-FOU ; 2020;17
Keywords [en]
Determinantal point processes, Periodically weighted random tilings, Central Limit Theorems, Riemann-Hilbert problems
National Category
Mathematical Analysis Probability Theory and Statistics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-273400ISBN: 978-91-7873-556-3 (print)OAI: oai:DiVA.org:kth-273400DiVA, id: diva2:1430679
Public defence
2020-06-12, Zoom, 10:00 (English)
Opponent
Supervisors
Available from: 2020-05-19 Created: 2020-05-16 Last updated: 2020-05-19Bibliographically approved
List of papers
1. Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble
Open this publication in new window or tab >>Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble
2017 (English)In: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 20, no 3, article id 19Article in journal (Refereed) Published
Abstract [en]

In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the limiting fluctuations are no longer Gaussian, but described by infinitely divisible laws. We argue that this transition phenomenon is universal by showing that the same transition and their laws appear for fluctuations of the thinned sine process in a growing box. The proofs are based on a Riemann-Hilbert problem for integrable operators.

Place, publisher, year, edition, pages
SPRINGER, 2017
Keywords
Random matrices, Central Limit Theorems, Linear statistics, Mesoscopic scales, Riemann-Hilbert problems
National Category
Mathematics Physical Sciences
Identifiers
urn:nbn:se:kth:diva-210963 (URN)10.1007/s11040-017-9250-4 (DOI)000404329900001 ()2-s2.0-85021200344 (Scopus ID)
Funder
Knut and Alice Wallenberg FoundationSwedish Research Council
Note

QC 20170712

Available from: 2017-07-12 Created: 2017-07-12 Last updated: 2020-05-16Bibliographically approved
2. Correlation functions for determinantal processes defined by infinite block Toeplitz minors
Open this publication in new window or tab >>Correlation functions for determinantal processes defined by infinite block Toeplitz minors
2019 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 356, article id 106766Article in journal (Refereed) Published
Abstract [en]

We study the correlation functions for determinantal point processes defined by products of infinite minors of block Toeplitz matrices. The motivation for studying such processes comes from doubly periodically weighted tilings of planar domains, such as the two-periodic Aztec diamond. Our main results are double integral formulas for the correlation kernels. In general, the integrand is a matrix-valued function built out of a factorization of the matrix-valued weight. In concrete examples the factorization can be worked out in detail and we obtain explicit integrands. In particular, we find an alternative proof for a formula for the two-periodic Aztec diamond recently derived in [20]. We strongly believe that also in other concrete cases the double integral formulas are good starting points for asymptotic studies.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2019
Keywords
Determinantal point processes, Non-negative block Toeplitz minors, Non-intersecting paths, Periodically weighted random tilings
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-263330 (URN)10.1016/j.aim.2019.106766 (DOI)000491211800001 ()2-s2.0-85071414409 (Scopus ID)
Note

QC 20191206

Available from: 2019-12-06 Created: 2019-12-06 Last updated: 2020-05-16Bibliographically approved
3. Domino tilings of the Aztec diamond with doubly periodic weightings
Open this publication in new window or tab >>Domino tilings of the Aztec diamond with doubly periodic weightings
(English)Manuscript (preprint) (Other academic)
Keywords
Determinantal point processes, Periodically weighted random tilings
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-273144 (URN)
Note

In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular a family of models which, for any k ∈ N, includes models with k smooth regions is analyzed as the size of the Aztec diamond tends to infinity. We use a non-intersecting paths formulation and give a double integral formula for the correlation kernel of the Aztec diamond of finite size. By a classical steepest descent analysis of the correlation kernel we obtain the local behavior in the smooth and rough regions as the size of the Aztec diamond tends to infinity. From the mentioned limit the macroscopic picture such as the arctic curves and in particular the number of smooth regions is deduced. Moreover we compute the limit of the height function and as a consequence we confirm, in the setting of this paper, that the limit in the rough region fulfills the complex Burgers’ equation, as stated by Kenyon and Okounkov.

Available from: 2020-05-07 Created: 2020-05-07 Last updated: 2020-05-16

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