Local universality for real roots of random trigonometric polynomials

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 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.