On global universality for zeros of random polynomials

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations ${\sum }_j{a}_j{P}_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j{\}}_j$ induced by a multi-circular weight function $Q$ satisfying suitable smoothness and growth conditions. In complex dimension $m\ge 3$, we prove that $\mathbb{E}[(\log (1+\mathrm{\mid }{a}_j\mathrm{\mid }){)}^m]<\mathrm{\infty }$ is a necessary and sufficient condition for normalized zero currents of random polynomials to be almost surely asymptotic to the (deterministic) extremal current $d{d}^c{V}_Q\mathrm{.}$ 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 similar results in this setting.