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A central limit theorem for fluctuations in polyanalytic ginibre ensembles
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-2041-0296
2017 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. rnx147Article in journal (Refereed) Epub ahead of print
Abstract [sv]

We study fluctuations of linear statistics in polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our main result is asymptotic normality of fluctuations, extending a result of Rider and Virág. As in the analytic case, the variance is composed of independent terms from the bulk and the boundary. Our methods rely on a structural formula for polyanalytic polynomial Bergman kernels which separates out the different pure q" role="presentation">q-analytic kernels corresponding to different Landau levels. The fluctuations with respect to these pure q" role="presentation">q-analytic Ginibre ensembles are also studied, and a central limit theorem is proved. The results suggest a stabilizing effect on the variance when the different Landau levels are combined together.

Place, publisher, year, edition, pages
2017. Vol. rnx147
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-228091DOI: 10.1093/imrn/rnx147OAI: oai:DiVA.org:kth-228091DiVA, id: diva2:1206740
Note

QC 20180518

Available from: 2018-05-17 Created: 2018-05-17 Last updated: 2018-05-18Bibliographically approved
In thesis
1. Random and optimal configurations in complex function theory
Open this publication in new window or tab >>Random and optimal configurations in complex function theory
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of six articles spanning over several areas of mathematical analysis. The dominant theme is the study of random point processes and optimal point configurations, which may be though of as systems of charged particles with mutual repulsion. We are predominantly occupied with questions of universality, a phenomenon that appears in the study of random complex systems where seemingly unrelated microscopic laws produce systems with striking similarities in various scaling limits. In particular, we obtain a complete asymptotic expansion of planar orthogonal polynomials with respect to exponentially varying weights, which yields universality for the microscopic boundary behavior in the random normal matrix (RNM) model (Paper A) as well as in the case of more general interfaces for Bergman kernels (Paper B). Still in the setting of RNM ensembles, we investigate properties of scaling limits near singular points of the boundary of the spectrum, including cusps points (Paper C). We also obtain a central limit theorem for fluctuations of linear statistics in the polyanalytic Ginibre ensemble, using a new representation of the polyanalytic correlation kernel in terms of algebraic differential operators acting on the classical Ginibre kernel (Paper D). Paper E is concerned with an extremal problem for analytic polynomials, which may heuristically be interpreted as an optimal packing problem for the corresponding zeros. The last article (Paper F) concerns a different theme, namely a sharp topological transition in an Lp-analogue of classical Carleman classes for 0 < p < 1.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2018. p. 46
Series
TRITA-SCI-FOU ; 2018:20
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-228092 (URN)978-91-7729-783-3 (ISBN)
Public defence
2018-06-08, Sal E2, Lindstedtsvägen 3, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20180518

Available from: 2018-05-18 Created: 2018-05-17 Last updated: 2018-05-18Bibliographically approved

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