On global universality for zeros of random polynomials

Authors : Turgay Bayraktar

Pages : 384-398

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Publication Date : 2019-04-01

Article Type : Research

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 $\mathbb{C}^m$ satisfying suitable smoothness and growth conditions. In complex dimension $m\geq3$, we prove that $\mathbb{E}[(\log(1+|a_j|))^m]<\infty$ 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.
Keywords : Random polynomial, distribution of zeros, global universality