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 $$f_n(t):=\frac{1}{\sqrt{n}}\sum _{k=1}^n{a}_k\cos (kt)+b_k\sin (kt),$$ where $(a_k{)}_{k\ge 1}$ and $(b_k{)}_{k\ge 1}$ are two independent stationary Gaussian processes with the same correlation function $\rho :k\mapsto \cos (k\alpha )$, with $\alpha \ge 0$. 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 $\epsilon >0$, for all $\mathrm{\ell }\in (\sqrt{2},2]$, there exists $\alpha \ge 0$ and $n\ge 1$ large enough such that $$\mid \frac{\mathbb{E}[\mathcal{N}(f_n,[0,2\pi ])]}{n}-\mathrm{\ell }\mid \le \epsilon ,$$ where $\mathcal{N}(f_n,[0,2\pi ])$ denotes the number of real zeros of the function $f_n$ in the interval $[0,2\pi ]$. 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.