Roots of Kostlan polynomials: moments, strong law of large numbers and central limit theorem. (Racines des polynômes de Kostlan: moments, loi forte des grands nombres et théorème central limite.) (English. French summary) Zbl 1487.60048
The authors study the number of real roots of a Kostlan random polynomial of degree \(d\) in one variable. These random polynomials are also known as elliptic polynomials in the literature [M. Sodin and B. Tsirelson, Isr. J. Math. 144, 125–149 (2004; Zbl 1072.60043)] The roots of such a polynomial form a random subset of \(\mathbb R\) denoted by \(Z_d.\) The authors consider the counting measure of the set of real roots of such polynomials and compute the large degree asymptotics of the central moments of these random variables. As a consequence, it is obtained 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.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)
MSC:
| 60F05 | Central limit and other weak theorems |
| 14P25 | Topology of real algebraic varieties |
| 32L05 | Holomorphic bundles and generalizations |
| 60F15 | Strong limit theorems |
| 60G15 | Gaussian processes |
| 60G57 | Random measures |
Keywords:
Kostlan polynomials; complex Fubini-study model; Kac-Rice formula; law of large numbers; central limit theorem; method of momentsCitations:
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