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Asymptotic normality of linear statistics of zeros of random polynomials


Author: Turgay Bayraktar
Journal: Proc. Amer. Math. Soc. 145 (2017), 2917-2929
MSC (2010): Primary 32A60, 32A25; Secondary 60F05, 60D05
DOI: https://doi.org/10.1090/proc/13441
Published electronically: December 30, 2016
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Abstract: In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $ L^2$-space of polynomials endowed with varying measures of the form $ e^{-2n\varphi _n(z)}dz$ under suitable assumptions on the weight functions $ \varphi _n$.


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Additional Information

Turgay Bayraktar
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
Address at time of publication: Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey
Email: tbayrakt@syr.edu, tbayraktar@sabanciuniv.edu

DOI: https://doi.org/10.1090/proc/13441
Keywords: Central limit theorem, linear statistics, random polynomial, Bergman kernel asymptotics
Received by editor(s): March 2, 2016
Received by editor(s) in revised form: March 18, 2016, March 28, 2016, June 14, 2016, and August 2, 2016
Published electronically: December 30, 2016
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society