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

Ancona, Michele; Letendre, Thomas

Annales Henri Lebesgue, Volume 4 (2021), pp. 1659-1703.

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.

Nous étudions le nombre de racines réelles d’un polynôme de Kostlan de degré $d$ en une variable. Plus généralement, nous nous intéressons à la distribution de la mesure de comptage de l’ensemble des racines réelles d’un tel polynôme. Nous obtenons l’asymptotique des moments centrés de ces variables aléatoires, dans la limite des grands degrés. Nous en déduisons une loi forte des grands nombres et un théorème central limite. En particulier, les racines réelles d’un polynôme de Kostlan s’équidistribuent presque surement lorsque le degré tend vers l’infini. De plus, les fluctuations de la mesure de comptage de cet ensemble aléatoire convergent en distribution vers le bruit blanc gaussien standard. Nos résultats sont valables plus généralement pour les zéros réels d’une section réelle aléatoire dans le modèle dit de Fubini–Study complexe.

Classification: 14P25, 32L05, 60F05, 60F15, 60G15, 60G57
Mots-clés : Kostlan polynomials, Complex Fubini–Study model, Kac–Rice formula, Law of Large Numbers, Central Limit Theorem, Method of moments.

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