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Mesoscopic fluctuations for unitary invariant ensembles
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2018 (English)In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 23, article id 7Article in journal (Refereed) Published
Abstract [en]

Considering a determinantal point process on the real line, we establish a connection between the sine-kernel asymptotics for the correlation kernel and the CLT for mesoscopic linear statistics. This implies universality of mesoscopic fluctuations for a large class of unitary invariant Hermitian ensembles. In particular, this shows that the support of the equilibrium measure need not be connected in order to see Gaussian fluctuations at mesoscopic scales. Our proof is based on the cumulants computations introduced in [45] for the CUE and the sine process and the asymptotic formulae derived by Deift et al. [13]. For varying weights e(-N) (Tr) (V) ((H)), in the one-cut regime, we also provide estimates for the variance of linear statistics Tr f (H) which are valid for a rather general function f. In particular, this implies that the logarithm of the absolute value of the characteristic polynomials of such Hermitian random matrices converges in a suitable regime to a regularized fractional Brownian motion with logarithmic correlations introduced in [17]. For the GUE and Jacobi ensembles, we also discuss how to obtain the necessary sine-kernel asymptotics at mesoscopic scale by elementary means.

Place, publisher, year, edition, pages
UNIV WASHINGTON, DEPT MATHEMATICS , 2018. Vol. 23, article id 7
Keywords [en]
unitary invariant ensembles, asymptotics of Christoffel-Darboux kernels, central limit theorem, universality, sine process
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-224064DOI: 10.1214/17-EJP120ISI: 000425368400003Scopus ID: 2-s2.0-85042235582OAI: oai:DiVA.org:kth-224064DiVA, id: diva2:1190756
Note

QC 20180315

Available from: 2018-03-15 Created: 2018-03-15 Last updated: 2018-03-15Bibliographically approved

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