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 ).