On global universality for zeros of random polynomials

TL;DR

This paper investigates the asymptotic distribution of zeros of random multivariable polynomials with i.i.d. coefficients, establishing conditions under which their zero currents converge to a deterministic extremal current, extending results to various dimensions.

Contribution

It provides a necessary and sufficient condition for zero current convergence in higher dimensions and extends the analysis to one-dimensional cases with regular measures.

Findings

Zero currents almost surely converge to the extremal current under the specified condition.

The condition involves the finiteness of the m-th moment of the log of the coefficients.

Results unify the understanding of zero distribution for random polynomials across different complex dimensions.

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 $Q$ 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 $dd^cV_{Q}.$ 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 similar results in this setting.