Zeros of real random polynomials spanned by OPUC

Let $\{{\phi }_i{\}}_{i=0}^{\mathrm{\infty }}$ be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure $\mu$. We study zero distribution of random linear combinations of the form $$P_n(z)=\sum _{i=0}^{n-1}{\eta }_i{\phi }_i(z),$$ where ${\eta }_0,\dots ,{\eta }_{n-1}$ are i.i.d. standard Gaussian variables. We use the Christoffel-Darboux formula to simplify the density functions provided by Vanderbei for the expected number real and complex of zeros of $P_n$. From these expressions, under the assumption that $\mu$ is in the Nevai class, we deduce the limiting value of these density functions away from the unit circle. Under the mere assumption that $\mu$ is doubling on subarcs of $\text{T}$ centered at $1$ and $-1$, we show that the expected number of real zeros of $P_n$ is at most $$(2\mathrm{/}\pi )\log n+O(1),$$ and that the asymptotic equality holds when the corresponding recurrence coefficients decay no slower than $n^{-(3+ϵ)\mathrm{/}2}$, $ϵ>0$. We conclude with providing results that estimate the expected number of complex zeros of $P_n$ in shrinking neighborhoods of compact subsets of $\text{T}$.