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Fluctuations of smooth linear statistics of determinantal point processes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We study eigenvalues of unitary invariant random matrices and other de-terminantal point processes. Paper A investigates some generalizations ofthe Gaussian Unitary Ensemble which are motivated by the physics of freefermions. We show that these processes exhibit a transition from Poisson tosine statistics at mesoscopic scales and that, at the critical scale, fluctuationsare not Gaussian but are governed by complicated limit laws. In papers Band C, we prove limit theorems which cover the different regimes of randommatrix theory. In particular, this establishes universality of the fluctuations ofinvariant Hermitian random matrices in great generality. The techniques arebased on generalizations of the orthogonal polynomial method and the cumu-lant method developed by Soshnikov. In particular, the results rely on certaincombinatorial identities originating in the theory of random walks and on theasymptotics for Orthogonal polynomials coming from the Riemann-Hilbertsteepest descent introduced by Deift et al.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2016. , x, 43 p.
Series
TRITA-MAT-A, 2016-1
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-186536OAI: oai:DiVA.org:kth-186536DiVA: diva2:927684
Public defence
2016-06-03, Sal F3,, Lindstedtsvägen 26, KTH-Campus, Stockholm, 13:15 (English)
Opponent
Supervisors
Funder
Knut and Alice Wallenberg Foundation, 67394
Note

QC 20160513

Available from: 2016-05-13 Created: 2016-05-12 Last updated: 2016-05-13Bibliographically approved
List of papers
1. Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
Open this publication in new window or tab >>Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186533 (URN)
Funder
Knut and Alice Wallenberg Foundation, 67394
Note

QC 20160513

Available from: 2016-05-12 Created: 2016-05-12 Last updated: 2016-05-13Bibliographically approved
2. Central limit theorem for biorthogonal ensembles and related combinatorial identities
Open this publication in new window or tab >>Central limit theorem for biorthogonal ensembles and related combinatorial identities
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186535 (URN)
Funder
Knut and Alice Wallenberg Foundation, 67394
Note

QC 20160513

Available from: 2016-05-12 Created: 2016-05-12 Last updated: 2016-05-13Bibliographically approved
3. Mesoscopic fluctuations for unitary invariant ensembles
Open this publication in new window or tab >>Mesoscopic fluctuations for unitary invariant ensembles
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186534 (URN)
Funder
Knut and Alice Wallenberg Foundation, 67394
Note

QCR 20160513

Available from: 2016-05-12 Created: 2016-05-12 Last updated: 2016-05-13Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
  • harvard1
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Language
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  • en-GB
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  • Other locale
More languages
Output format
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