Roots of Kostlan polynomials: moments, strong Law of Large Numbers and Central Limit Theorem

TL;DR

This paper analyzes the distribution of real roots of Kostlan random polynomials, establishing asymptotic moments, a strong Law of Large Numbers, and a Central Limit Theorem, showing roots become equidistributed with Gaussian fluctuations as degree grows.

Contribution

It provides the first detailed asymptotic analysis of moments and distributional limits for the real roots of Kostlan polynomials, including a CLT and equidistribution results.

Findings

Real roots almost surely equidistribute as degree increases

Moments of root counts follow specific asymptotics

Fluctuations converge to Gaussian White Noise

Abstract

We study the number of real roots of a Kostlan random polynomial of degree $d$ in one variable. More generally, we are interested in the distribution of the counting measure of the set of real roots of such a polynomial. We compute the asymptotics of the central moments of any order of these random variables, in the large degree limit. As a consequence, we prove that these quantities satisfy a strong Law of Large Numbers and a Central Limit Theorem. In particular, the real roots of a Kostlan polynomial almost surely equidistribute as the degree diverges. Moreover, the fluctuations of the counting measure of this random set around its mean converge in distribution to the Gaussian White Noise. More generally, our results hold for the real zeros of a random real section of a line bundle of degree d over a real projective curve, in the complex Fubini--Study model.