# On global universality for zeros of random polynomials

Bayraktar, Turgay (2017) On global universality for zeros of random polynomials. Hacettepe Journal of Mathematics and Statistics . ISSN 1303-5010 Published Online First http://www.hjms.hacettepe.edu.tr/Early-Access

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## Abstract

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by a multi-circular weight function $Q$ defined on $\C^m$ satisfying suitable smoothness and growth conditions. In complex dimension $m\geq3$, we prove that $\Bbb{E}[(\log(1+|a_j|))^m]<\infty$ is a necessary and sufficient condition for normalized zero currents of random polynomials to be almost surely asymptotic to the (deterministic) extremal current $\frac{i}{\pi}\partial\overline{\partial}V_{Q}.$ In addition, in complex dimension one, we consider random linear combinations of orthonormal polynomials with respect to a regular measure in the sense of Stahl \& Totik and we prove analogous results in this setting.

Item Type: Article Q Science > QA Mathematics > QA299.6-433 Analysis 35477 Turgay Bayraktar 11 Aug 2018 15:27 11 Aug 2018 15:27

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