Random Sums of Weighted Orthogonal Polynomials in ${\mathbb C}^d$

TL;DR

This paper studies the zeros of random polynomials in several complex variables, showing they converge to a deterministic measure under certain conditions, extending classical results to higher dimensions.

Contribution

It establishes convergence of zero measures for random weighted orthogonal polynomials in multiple complex variables, generalizing known one-dimensional results.

Findings

Zero measures converge to weighted equilibrium measures in dimension 1

Zero currents converge in higher dimensions

Optimal moment conditions are identified for convergence

Abstract

We consider random polynomials of the form $G_n(z):= \sum_{|\alpha|\leq n} \xi^{(n)}_{\alpha}p_{n,\alpha}(z)$ where $\{\xi^{(n)}_{\alpha}\}_{|\alpha|\leq n}$ are i.i.d. (complex) random variables and $\{p_{n,\alpha}\}_{|\alpha|\leq n}$ form a basis for $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${\mathbb C}^d$. In particular, this includes the setting where $\{p_{n,\alpha}\}$ are orthonormal in a space $L^2(e^{-2n Q} \tau)$, where $\tau$ is a compactly supported Bernstein-Markov measure and $Q$ is a continuous weight function. Under an optimal moment condition on the random variables $\{\xi^{(n)}_{\alpha}\}$, in dimension $d=1$ we prove convergence in probability of the zero measure to the weighted equilibrium measure, and in dimension $d \ge 2$ we prove convergence of zero currents.