Title: Zeros of real random polynomials spanned by OPUC,
Abstract: Let \( \{\varphi_i\}_{i=0}^\infty \) 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 \[
P_n(z)=\sum_{i=0}^{n-1}\eta_i\varphi_i(z), \] 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 \( \T \) centered at
\( 1 \) and \( -1 \), we show that the expected number of real zeros of \( P_n
\) is at most \[ (2/\pi) \log n +O(1), \] and that the asymptotic equality
holds when the corresponding recurrence coefficients decay no slower than \(
n^{-(3+\epsilon)/2} \), \( \epsilon>0 \). We conclude with providing results
that estimate the expected number of complex zeros of \( P_n \) in shrinking
neighborhoods of compact subsets of \( \T \).