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

Abstract

Let $$\{\varphi _k\}_{k=0}^\infty $$ be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure $$ \mu $$. We study the variance of the number of zeros of random linear combinations of the form $$\begin{aligned} P_n(z)=\sum _{k=0}^{n}\eta _k\varphi _k(z), \end{aligned}$$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 $$v(\theta )\ge c>0$$, $$\theta ,\theta _j\in [0,2\pi )$$, and $$\alpha _j>0$$, we give a quantitative estimate on 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 obtain 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.