Central Limit Theorem for the number of real roots of random orthogonal polynomials

TL;DR

This paper proves that the number of real roots of a broad class of Gaussian random orthogonal polynomials follows a normal distribution asymptotically, despite varying local correlations.

Contribution

It establishes a central limit theorem for the count of real roots in random orthogonal polynomials using Wiener Chaos methods.

Findings

Number of real roots is asymptotically Gaussian in the bulk.

Fluctuations are normal even with non-uniform local correlations.

Applicable to a wide class of Gaussian orthogonal polynomials.

Abstract

In this note we study the number of real roots of a wide class of random orthogonal polynomials with gaussian coefficients. Using the method of Wiener Chaos we show that the fluctuation in the bulk is asymptotically gaussian, even when the local correlations are not necessarily the same.