Zeros of random linear combinations of OPUC with complex Gaussian coefficients

TL;DR

This paper analyzes the distribution of zeros of random linear combinations of orthogonal polynomials on the unit circle with complex Gaussian coefficients, deriving explicit formulas and limits for their zero densities.

Contribution

It provides explicit intensity functions for zero distributions of these random polynomials and explores their limits for Szegő weights.

Findings

Derived explicit zero intensity functions using Christoffel-Darboux formula.

Established the limiting behavior of zero densities for Szegő weights.

Provided simple formulas for zero distribution in Jordan regions.

Abstract

We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^n\eta_j\phi_j(z),$$ in any Jordan region $\Omega \subset \mathbb C$. The basis functions $\phi_j$ are orthogonal polynomials on the unit circle (OPUC) that are real-valued on the real line, and $\eta_0,\dots,\eta_n$ are complex-valued iid Gaussian random variables. We derive an explicit intensity function for the number of zeros of $P_n$ in $\Omega$ for each fixed $n$. Using the Christoffel-Darboux formula, the intensity function takes a very simple shape. Moreover, we give the limiting value of the intensity function when the orthogonal polynomials are associated to Szeg\H{o} weights.