An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

Abstract

Let {φi}i=0∞ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say En(μ), of random polynomialsPn(z):=∑i=0nηiφi(z), where η0,…,ηn are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that En(|dξ|) admits an asymptotic expansion of the formEn(|dξ|)∼2πlog⁡(n+1)+∑p=0∞Ap(n+1)−p (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon–Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case En(μ) admits an analogous expansion with the coefficients Ap depending on the measure μ for p≥1 (the leading order term and A0 remain the same).