The distribution of the zeroes of random trigonometric polynomials

TL;DR

This paper investigates the asymptotic distribution of zeros of random trigonometric polynomials, proving Gaussian convergence for the normalized zero count and extending results to short intervals.

Contribution

It establishes the Gaussian limit for the normalized number of zeros of random trigonometric polynomials and confirms the predicted variance growth rate.

Findings

Normalized zero count converges to Gaussian distribution.

Variance of zero count grows linearly with degree.

Results extend to zeros in short intervals.

Abstract

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. 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 $\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.