Roots of Kostlan polynomials: moments, strong Law of Large Numbers and Central Limit Theorem
Annales Henri Lebesgue, Volume 4 (2021), pp. 1659-1703.

KeywordsKostlan polynomials, Complex Fubini–Study model, Kac–Rice formula, Law of Large Numbers, Central Limit Theorem, Method of moments.

### Abstract

We study the number of real roots of a Kostlan random polynomial of degree $d$ in one variable. More generally, we consider the counting measure of the set of real roots of such polynomials. We compute the large degree asymptotics of the central moments of these random variables. As a consequence, we obtain 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 their counting measure converge in distribution to the Standard Gaussian White Noise. More generally, our results hold for the real zeros of a random real section in the complex Fubini–Study model.

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