Random orthonormal polynomials: local universality and expected number of real roots

TL;DR

This paper proves universal behavior in the number of real roots of random orthonormal polynomials with broad classes of coefficient distributions, extending previous Gaussian-only results to more general cases.

Contribution

It establishes universality of the asymptotic number of real roots for a wide class of random orthonormal polynomials beyond Gaussian coefficients.

Findings

Universal asymptotics for the number of real roots

Local and global root distribution results

Extension beyond Gaussian coefficient assumptions

Abstract

We consider random orthonormal polynomials $$ F_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, \dots, $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep)$ moments, and $(p_n)_{n=0}^{\infty}$ is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of $F_n$, both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients \cite{lubinsky2016linear} using the Kac-Rice formula, a method that does not extend to the generality of our paper.