Expected number of real zeros for random orthogonal polynomials

TL;DR

This paper investigates the expected number of real zeros in random orthogonal polynomials, revealing asymptotic behaviors and distributional limits, extending known results from classical polynomial bases to a broad class of weights.

Contribution

It generalizes the asymptotic expected number of real zeros for random orthogonal polynomials beyond classical cases, including local zero distribution results.

Findings

Expected zeros asymptotically proportional to n/√3 for broad weight classes

Weak convergence of scaled zero counting measures to Ullman or arcsine distributions

Extension of classical results to a large class of weights on the real line

Abstract

We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/\pi + o(1))\log{n}$ expected real zeros in terms of the degree $n$. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight defined on the whole real line, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution.