Zeros of real random polynomials spanned by OPUC

TL;DR

This paper analyzes the distribution of zeros of random polynomials formed from orthonormal polynomials on the unit circle, deriving asymptotic behaviors and bounds for real and complex zeros under various measure conditions.

Contribution

It provides new formulas for zero densities of random OPUC-based polynomials and establishes asymptotic zero counts under Nevai class and doubling measure assumptions.

Findings

Expected number of real zeros is at most (2/π) log n + O(1).

Asymptotic equality for real zeros when recurrence coefficients decay sufficiently.

Derived estimates for the expected number of complex zeros near the unit circle.

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 \( \T \).