Variance of real zeros of random orthogonal polynomials

TL;DR

This paper analyzes the asymptotic behavior of the variance in the number of real zeros of random orthogonal polynomials, showing it grows linearly with degree and providing explicit constants.

Contribution

It provides the first detailed asymptotic analysis of the variance of zeros for random orthogonal polynomials, including explicit constants.

Findings

Variance asymptotically proportional to degree n

Explicit constant c for variance growth

Asymptotic behavior holds for subintervals of the support

Abstract

We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree $\leq n$ in subintervals $\left [ a,b\right ] $ of the support of the underlying orthogonality measure $\mu $. We show that, as $n\rightarrow \infty $, this variance is asymptotic to $cn$, for some explicit constant $c>0$.