Local universality for real roots of random trigonometric polynomials

TL;DR

This paper proves local universality for the distribution of real roots of certain random trigonometric polynomials, showing convergence to a Gaussian process with a specific correlation function, regardless of the underlying distribution of coefficients.

Contribution

It establishes that the local distribution of zeros of these polynomials converges to that of a universal Gaussian process, extending to various coefficient distributions.

Findings

Zeros converge to those of a Gaussian process with correlation (\

")/t

Universality holds for coefficients with arbitrary covariance or in the domain of attraction of a stable law

Abstract

Consider a random trigonometric polynomial $X_n: \mathbb R \to \mathbb R$ of the form $$ X_n(t) = \sum_{k=1}^n \left( \xi_k \sin (kt) + \eta_k \cos (kt)\right), $$ where $(\xi_1,\eta_1),(\xi_2,\eta_2),\ldots$ are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\in\mathbb N}$ be any sequence of real numbers. We prove that as $n\to\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian process with correlation function $(\sin t)/t$. We also establish similar local universality results for the centered random vectors $(\xi_k,\eta_k)$ having an arbitrary covariance matrix or belonging to the domain of attraction of a two-dimensional $\alpha$-stable law.