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

TL;DR

This paper derives an asymptotic expansion for the expected number of real zeros of random polynomials spanned by orthogonal polynomials on the unit circle, generalizing previous results for Kac polynomials.

Contribution

It extends Wilkins' asymptotic expansion for the expected zeros to a broader class of measures with holomorphic Radon-Nikodym derivatives.

Findings

Asymptotic expansion of expected zeros includes a logarithmic leading term.

Coefficients depend on the measure's properties.

Generalization from arc length measure to more general measures.

Abstract

Let $ \{\varphi_i\}_{i=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure $ \mu $ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(\mu) $, of random polynomials \[ P_n(z) := \sum_{i=0}^n\eta_i\varphi_i(z), \] where $ \eta_0,\dots,\eta_n $ are i.i.d. standard Gaussian random variables. When $ \mu $ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that $ \mathbb E_n(|\mathrm d\xi|) $ admits an asymptotic expansion of the form \[ \mathbb E_n(|\mathrm d\xi|) \sim \frac2\pi\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where $ \mu $ 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 $ \mathbb E_n(\mu) $ admits an analogous expansion with coefficients the $ A_p $ depending on the measure $ \mu $ for $ p\geq 1 $ (the leading order term and $ A_0 $ remain the same).