Roots of random trigonometric polynomials with general dependent coefficients

TL;DR

This paper demonstrates that the expected number of real zeros in large-degree random trigonometric polynomials remains consistent with the independent case, even when coefficients are dependent, highlighting a universality in their behavior.

Contribution

It establishes the first universality result for the expected number of real zeros in non-Gaussian dependent coefficient settings.

Findings

Expected zeros asymptotics match independent case

Universality holds under mild dependence hypotheses

Robustness of zeros despite dependencies

Abstract

We consider random trigonometric polynomials with general dependent coefficients. We show that under mild hypotheses on the structure of dependence, the asymptotics as the degree goes to infinity of the expected number of real zeros coincides with the independent case. To the best of our knowledge, this universality result is the first obtained in a non-Gaussian dependent context. Our proof highlights the robustness of real zeros, even in the presence of dependencies. These findings bring the behavior of random polynomials closer to real-world models, where dependencies between coefficients are common.