New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients

We consider random trigonometric polynomials of the form $f_n(t):=\frac{1}{\sqrt{n}} \sum_{k=1}^{n}a_k \cos(k t)+b_k \sin(k t),$ where $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent stationary Gaussian processes with the same correlation function $ρ: k \mapsto \cos(kα)$, with $α\geq 0$. We show that the asymptotics of the expected number of real zeros differ from the universal one $\frac{2}{\sqrt{3}}$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $\varepsilon>0$, for all $\ell \in (\sqrt{2},2]$, there exists $α\geq 0$ and $n\geq 1$ large enough such that $\left|\frac{\mathbb{E}\left[\mathcal{N}(f_n,[0,2π])\right]}{n}-\ell\right|\leq \varepsilon,$ where $\mathcal N(f_n,[0,2π])$ denotes the number of real zeros of the function $f_n$ in the interval $[0,2π]$. Therefore, this result provides the first example where the expected number of real zeros do not converge as $n$ goes to infinity by exhibiting a whole range of possible limits ranging from $\sqrt{2}$ to 2.

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