Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients

TL;DR

This paper proves that the asymptotic distribution of the number of zeros of random trigonometric polynomials is universal, depending only on the mean and variance of coefficients, not their specific distribution, under certain conditions.

Contribution

It establishes a universality result showing the zero count distribution converges to that of a Gaussian process, regardless of the coefficient distribution, given certain density conditions.

Findings

Zero count distribution converges to a universal limit.

Universality holds for coefficients with certain density conditions.

Results extend previous Gaussian-specific asymptotics.

Abstract

Let $X_N$ be a random trigonometric polynomial of degree $N$ with iid coefficients and let $Z_N(I)$ denote the (random) number of its zeros lying in the compact interval $I\subset\mathbb{R}$. Recently, a number of important advances were made in the understanding of the asymptotic behaviour of $Z_N(I)$ as $N\to\infty$, in the case of standard Gaussian coefficients. The main theorem of the present paper is a universality result, that states that the limit of $Z_N(I)$ does not really depend on the exact distribution of the coefficients of $X_N$. More precisely, assuming that these latter are iid with mean zero and unit variance and have a density satisfying certain conditions, we show that $Z_N(I)$ converges in distribution toward $Z(I)$, the number of zeros within $I$ of the centered stationary Gaussian process admitting the cardinal sine for covariance function.