# The Variance of the Number of Zeros for Complex Random Polynomials Spanned by OPUC

## TL;DR

This paper investigates the variance of zeros of random linear combinations of orthonormal polynomials on the unit circle, providing estimates and limiting formulas under various conditions on the polynomials and random coefficients.

## Contribution

It offers new quantitative estimates and formulas for the variance of zeros of complex random polynomials spanned by OPUC, extending understanding in different regularity and measure cases.

## Key findings

- Variance estimates for zeros in sectors intersecting the unit circle.
- Limiting variance formulas for zeros in annuli away from the unit circle.
- Results apply to polynomials with coefficients satisfying certain moment bounds.

## Abstract

Let $\{\varphi_k\}_{k=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ \mu $. We study the variance of the number of zeros of random linear combinations of the form $$ P_n(z)=\sum_{k=0}^{n}\eta_k\varphi_k(z), $$ where $\{\eta_k\}_{k=0}^n $ are complex-valued random variables. Under the assumption that the distribution for each $\eta_k$ satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when $\{\varphi_k\}$ are regular in the sense of Ullman-Stahl-Totik or are such that the measure of orthogonality $\mu$ satisfies $d\mu(\theta)=w(\theta)d\theta$ where $w(\theta)=v(\theta)\prod_{j=1}^J|\theta - \theta_j|^{\alpha_j}$, with $v(\theta)\geq c>0$, $\theta,\theta_j\in [0,2\pi)$, and $\alpha_j>0$, we give a quantitative estimate on the the variance of the number of zeros of $P_n$ in sectors that intersect the unit circle. When $\{\varphi_k\}$ are real-valued on the real-line from the Nevai class and $\{\eta_k\}$ are i.i.d.~complex-valued standard Gaussian, we prove a formula for the limiting value of variance of the number of zeros of $P_n$ in annuli that do not contain the unit circle.