# On global universality for zeros of random polynomials

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Bayraktar, Turgay (2019) On global universality for zeros of random polynomials. Hacettepe Journal of Mathematics and Statistics, 48 (2). pp. 384-398. ISSN 1303-5010

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Official URL: http://dx.doi.org/10.15672/HJMS.2017.525

## 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 Random polynomial; distribution of zeros; global universality Q Science > QA Mathematics > QA299.6-433 Analysis Q Science > QA Mathematics 36981 Turgay Bayraktar 03 May 2019 11:19 22 May 2019 14:14

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