On global universality for zeros of random polynomials

Abstract

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 $\mathbb{E}[(\log (1+|a_j|){)}^m]<\infty$ is a necessary and sufficient condition for normalized zero currents of random polynomials to be almost surely asymptotic to the (deterministic) extremal current $d{d}^c{V}_Q.$ 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 similar results in this setting.

15 pages

References in corpus (4)

• Asymptotic distribution of complex zeros of random analytic functions
• Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
• Equilibrium distribution of zeros of random polynomials
• Expected number of real roots for random linear combinations of orthogonal polynomials associated with radial weights

Cited by in corpus (3)

• A necessary and sufficient condition for convergence of the zeros of random polynomials
• Mass equidistribution for random polynomials
• A survey on zeros of random holomorphic sections