Abstract
We study the number of real roots of a Kostlan (or elliptic) 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. We also prove that the random variables we study concentrate in probability around their mean faster than any negative power of d. 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.
Highlights
Kostlan polynomialsA real Kostlan polynomial of degree d is a univariate random polynomial of the form d ak k=0 d Xk, k where the coefficients[0] k d are independent N (0, 1) random variables
We prove a strong Law of Large Numbers (Theorem 1.7) and a Central Limit Theorem (Theorem 1.9) for the linear statistics of the random measuresd ∈ N defined above
Kac–Rice formulas are a classical tool in the study of the number of real roots of random polynomials
Summary
A real Kostlan polynomial of degree d is a univariate random polynomial of the form d ak k=0 d Xk, k where the coefficients (ak)[0] k d are independent N (0, 1) random variables. Where σ is the constant appearing in Dalmao’s variance estimate, and (μp)p ∈ N is the sequence of moments of the standard real Gaussian distribution N (0, 1). This results allows us to prove a strong. We denote by Zd = Zsd and by νd = νsd for simplicity In this setting, the linear statistics of νd were studied in [Anc[21], GW16, Let[16], LP19], among others. In [Anc21], Ancona derived a two terms asymptotic expansion of the non-central moments of the linear statistics of νd As a consequence, he proved the following (cf [Anc[21], Theorem 0.5]). All the results in [Anc21] are still valid for random real sections of E ⊗ Ld → X
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