The distribution of the zeros of random trigonometric polynomials

By Andrew Granville, I. Wigman. 2008

65 citations recorded by the source corpus.

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\rightarrow\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that $\frac{Z_{N}-{\Bbb E} Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9738fe47031efc03ac870326ff37e7c20fd5176b

Citations in the corpus

• Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d (2014, unknown): “≡ − the weight function w ( t ) = (1 − t ) (cid:12) In order to formulate our main result, let us introduce the usual Wasserstein distance d between random variables Z , N : where Lip(1) is the set of Lipschitz functions whose Lipschitz constant equals 1.”