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

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.