LAUSR

Let $$\{\varphi _k\}_{k=0}^\infty $$ { φ k } k = 0 ∞ be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure $$ \mu $$ μ . We study the variance of the number of zeros of random linear combinations of the form $$\begin{aligned} P_n(z)=\sum _{k=0}^{n}\eta _k\varphi _k(z), \end{aligned}$$ P n ( z ) = ∑ k = 0 n η k φ k ( z ) , where $$\{\eta _k\}_{k=0}^n $$ { η k } k = 0 n are complex-valued random variables. Under the assumption that the distribution for each $$\eta _k$$ η k satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when $$\{\varphi _k\}$$ { φ k } are regular in the sense of Ullman–Stahl–Totik or are such that the measure of orthogonality $$\mu $$ μ satisfies $$d\mu (\theta )=w(\theta )d\theta $$ d μ ( θ ) = w ( θ ) d θ where $$w(\theta )=v(\theta )\prod _{j=1}^J|\theta - \theta _j|^{\alpha _j}$$ w ( θ ) = v ( θ ) ∏ j = 1 J | θ - θ j | α j , with $$v(\theta )\ge c>0$$ v ( θ ) ≥ c > 0 , $$\theta ,\theta _j\in [0,2\pi )$$ θ , θ j ∈ [ 0 , 2 π ) , and $$\alpha _j>0$$ α j > 0 , we give a quantitative estimate on the variance of the number of zeros of $$P_n$$ P n in sectors that intersect the unit circle. When $$\{\varphi _k\}$$ { φ k } are real-valued on the real-line from the Nevai class and $$\{\eta _k\}$$ { η 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 $$P_n$$ P n in annuli that do not contain the unit circle.