Fluctuations of smooth linear statistics of determinantal point processes
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
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.
Research subject Mathematics
IdentifiersURN: urn:nbn:se:kth:diva-186536OAI: oai:DiVA.org:kth-186536DiVA: diva2:927684
2016-06-03, Sal F3,, Lindstedtsvägen 26, KTH-Campus, Stockholm, 13:15 (English)
Krasovsky, Igor, Dr.
FunderKnut and Alice Wallenberg Foundation, 67394
QC 201605132016-05-132016-05-122016-05-13Bibliographically approved
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