Zeros of random orthogonal polynomials with complex Gaussian coefficients

TL;DR

This paper analyzes the distribution of zeros of random linear combinations of orthogonal polynomials with complex Gaussian coefficients, revealing clustering near the unit circle and providing explicit density formulas.

Contribution

It derives explicit formulas for zero density of random orthogonal polynomials and characterizes their asymptotic zero distribution under minimal assumptions.

Findings

Zeros cluster near the unit circle for orthogonal polynomials on the circle

Explicit density functions for expected zero counts are obtained

Asymptotic zero distribution is characterized away from the orthogonality set

Abstract

Let $\{f_j\}_{j=0}^n$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^n\eta_jf_j(z),$$ where $\eta_0,\dots,\eta_n$ are complex-valued i.i.d.~standard Gaussian random variables. Using the Christoffel-Darboux formula, the density function for the expected number of zeros of $P_n$ in these cases takes a very simple shape. From these expressions, under the mere assumption that the orthogonal polynomials are from the Nevai class, we give the limiting value of the density function away from their respective sets where the orthogonality holds. In the case when $\{f_j\}$ are orthogonal polynomials on the unit circle, the density function shows that the expected number of zeros of $P_n$ are clustering near the unit circle. To quantify this phenomenon, we give a result that estimates the expected number of complex zeros of $P_n$ in shrinking neighborhoods of compact subsets of the unit circle.