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 $\rho: k \mapsto \cos(k\alpha)$, with $\alpha\geq 0$. 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 $\alpha \geq 0$ and $n\geq 1$ large enough such that $$\left|\frac{\mathbb{E}\left[\mathcal{N}(f_n,[0,2\pi])\right]}{n}-\ell\right|\leq \varepsilon,$$ 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.

1 Introduction and statement of the results

Real zeros of random trigonometric polynomials
There is tremendous amount of literature about complex or real zeros of random polynomials and their asymptotics as the degree of the latter goes to infinity. Recently, the universality of these asymptotics has been established in a certain number of models, see e.g. [Kac43,IM68,Far86,Mat10,Muk18,NNV14,DNV18] in the case of algebraic polynomials and [AP15,ADL,Fla17,IKM16,ADP19] in the case of trigonometric polynomials. The notion of universality stands here for the fact that these asymptotics do not depend on the choice of the law of the random entries, and to a certain extent, nor their correlation.
For example, in the case of trigonometric polynomials, it was shown in the last references that the first order asymptotics of the expected number of real zeros is indeed universal under very mild assumptions on the random coefficients, e.g. even in the presence of an arbitrary long-range correlation. This naturally raises the question of the existence of choices of "exotic" random entries such that the asymptotics of the expected number of real zeros do not coincide with the universal one.
We address this question here by exhibiting, for the first time, a simple model of random trigonometric polynomials, whose average number of real zeros does not converge as their degree goes to infinity. Our model belongs to the large class random trigonometric polynomials of the form where (a k ) k≥1 and (b k ) k≥1 are two independent stationary Gaussian processes with correlation function ρ : IN → IR, namely E[a k a l ] = E[b k b l ] =: ρ(|k − l|) and E[a k b l ] = 0 for all k, l ≥ 1. Thanks to Bochner's theorem, we then know that ρ is given by the Fourier transform of a measure µ, called the spectral measure. The case where ρ(k) = 0 for all k ≥ 1 of course corresponds to independent Gaussian coefficients as first studied by Dunnage in [Dun66]. Latter, in [Sam78] and [RS84], the authors considered the two "extreme" cases where E[aiaj] = ρ0 ∈]0, 1[ and E[aiaj] = ρ |i−j| 0 respectively. More recently, the authors of [ADP19] considered the case where the spectral measure admits a density satisfying mild hypotheses. In all these cases, it was shown that N (fn, [0, 2π]), the number of real zeros of the random function fn in the interval [0, 2π], obeys the same limit lim n→+∞ E[N (fn, [0, 2π])] n = 2 √ 3 .
In fact, considering standard Gaussian coefficients, one way to obtain asymptotics that do not match the universal one 2/ √ 3 is to consider either palindromic entries as in [FL12] or very special pairwise block entries such as in Theorem 2.3 and 2.4 of [Pir19]. We consider here the natural and purely singular case where the spectral measure is given by µ := 1 2 (δα + δ−α), for some real α ≥ 0. In other words, we consider the cosine correlation function If α ∈ πQ, the correlation function is thus periodic and the corresponding random coefficients of fn are strongly correlated at arbitrary large distance. If α / ∈ πQ, the sequence (ρ(k)) k≥0 is dense in [−1, 1] and the correlations between the random coefficients of fn becomes really intricate. We shall see that the asymptotics of the number of real zeros of fn then heavily depends on the arithmetic nature of α and more precisely on the distance of nα to πZ.

Statement of our results
Naturally, since fn is a random trigonometric polynomial of degree n, its number of zeros in bounded by 2n. In the case where nα ∈ πZ, we show that the expected number of real zeros is maximal in the following sense.
Proposition 1.1. If α = 0, then for all n ≥ 1 we have almost surely The case nα / ∈ πZ is more intriguing: properly renormalized, the expected number of real zeros of fn does not converge as n goes to infinity and admits in fact a whole continuum of possible limits. To be more precise, let us introduce the function α : (0, π) → R + defined by In Section 3.1.1 below, we examine the properties of α and its pointwise limit as α goes to zero The main result of the paper is then the following one.
Theorem 1.1. For all 0 < β < 1 and for all n large enough such that nα / ∈ πZ, we have The above theorem shows that if n is sufficiently large but nα stays away enough from πZ, then the expected number of real zeros on fn divided by n is close to the value of the function α at the point nα mod π. In particular, if α ∈ πQ, then the sequence (nα mod π) n≥1 takes values in a finite set S. From the above Theorem 1.1, we can then deduce the following corollary.
Corollary 1.2. Let us fix x ∈ (0, π) and consider a increasing subsequence (ϕ(n)) n≥1 such that ϕ(n)α converges to x as n goes to infinity. Then as spectral measure.
Remark 1.1. For sake of clarity, we only deal here with a spectral measure µ with one atom α and its opposite −α, but the method employed will work for any finite combination of atoms (±αi)i.
The rest of the paper is devoted to the proofs of the results stated above. Namely, in the next Section 2, we give the proof of Proposition 1.1, starting from the very simple case α = 0 and then generalizing to the case where α ∈ πQ and nα ∈ πZ. The last Section 3 is devoted to the proof of the main Theorem 1.1 and its corollaries in the case where nα / ∈ πZ. In this case, the study of the number of zeros is split into to parts: in Section 3.1 we determine the number of zeros away from the atoms ±α of the spectral measure µ. Finally, the numbers of zeros in the neighborhood of the atoms is shown to be negligible in the last Section 3.2.
2 Asymptotics in the case nα ∈ πZ In this Section, we give the proof of Proposition 1.1 describing the asymptotics of the number of real zeros of fn under the condition nα ∈ πZ.

The case α = 0
Let us first consider the very particular case where α = 0 i.e. the correlation function ρ is constant equal to one.

The case α ∈ πQ and nα ∈ πZ
Let us now suppose that α = 2πp q for positive and coprime integers p and q, i.e. the correlation sequence (ρ(k)) k is q−periodic. In this case, if n = qr for some positive integer r, we have nα ∈ Z and fn admits the following factorization where we have set The above factorization of fn invites to distinguish deterministic and random zeros. We have n − q deterministic zeros given by sin nt 2 = 0 and sin qt 2 = 0 ⇐⇒ t ∈ 2kπ n , k ∈ {0, . . . , n − 1}, r k .
Therefore the second statement in Proposition 1.1 follows from the following result which implies that, in the above framework, the expected number of real zeros of fn is asymptotic to n.
Proposition 2.2. As n tends to infinity, we have Since qα ∈ πZ, we have thus for t ∈ [0, 2π] ] > 0 and applying Kac-Rice formula (see e.g. Theorem 3.2 p. 71 of [AW09]), we get A straightforward computation shows that as n goes to infinity, Injecting this estimate in equation (2), we deduce that as n goes to infinity Letting ε go to zero, we finally get that 3 Asymptotics in the case nα / ∈ πZ We now consider the more intriguing case where nα / ∈ πZ. Following [ADP19], the variance and covariance of (fn(t), f n (t)) can then be written as convolutions of the spectral measure µ with explicit trigonometric kernels, namely where is the Fejer kernel, so that the normalization constant αn is given by αn := 6/(n + 1)(2n + 1) and Then for all n ≥ 1 such that nε > 1, we have the uniform estimates Proof. The estimate for K n is immediate. Let us set , we get that as soon as nε > 1 In the case we consider here, the spectral measure µ is 1 2 (δα + δ−α) so that we have simply The Fejér kernel being non negative, for n ≥ 1, we have Under the assumption nα / ∈ πZ, the distribution of the Gaussian variable fn(t) is thus nondegenerated for all t ∈ [0, 2π] and as above, we can use Kac-Rice formula (see e.g. [AW09]) to compute the expectation of N (fn, [0, 2π]), namely where In(t) := 1 αn We split the computation of the integral into two parts, depending on the proximity between the integration variable t and the atoms ±α of the spectral measure µ.
In the same manner, we have . Now remark that uniformly on Jε we have Therefore, uniformly on Jε we get where In particular, we get .
In order to make explicit the asymptotics of the right hand side of the last equation, let us now introduce an auxilary function and detail some of its properties.

An auxilary function and its properties
Remark that u → g α x (s, u) is then 2π−periodic and that we have the identification The function (u, s) → g α x (s, u) has singularities at (s, u) = ±(α, x) but these sigularities are integrable in the following sense.
The function 0 appears naturally as the pointwise limit of α when α ∈ (0, π) goes to zero.
Let us now show that the second term in Equation (8) converges to zero as α goes to zero. By symmetry, we can restrict ourselves to the case s ∈ [0, ]. Recall that when ω is small enough, there exists some constants C1, C2 > 0 such that C1|ω| ≤ | sin(ω)| ≤ C2|ω|. Thus, there exists C > 0 such that Set δ > 0 small enough such that for all u ∈ [x−δ, x+δ], we have sin u−x 2 ≥ C δ |u−x| and sin u+x 2 ≥ C δ sin(x). Using the fact that s + α ≥ α, we get that for some the constant C which may change from line to line and using the fact that x → x arctan 1 x is bounded on the real line, we get The same method naturally works in the neighborhood of −x. Otherwise, if we denote by hence the result.
Let us conclude this section with some properties of the limit function 0 (x).

From Riemann sum to integral
We can now establish the asymptotics of Equation (4) as n goes to infinity. As a first step, the integral of interest admits the following lower and upper bounds.
Lemma 3.7. If nε >> 1, then as n goes to infinity, we have Proof. We give the proof of the upper bound, the lower bound can be treated in the exact same way. To simplify the expressions, let us set E k n := 2πk n , 2π(k+1) n for 0 ≤ k ≤ n − 1. We can then decompose the integral on Jε as n + u n , u 1 E k n ⊂J ε/2 du.
Using the estimate (7) of Lemma 3.4, one then deduces that (11) Using again Equation (7) of Lemma 3.4, for all 0 ≤ k ≤ n − 1 such that E k n ⊂ J ε/2 , we have uniformly in u Integrating in u, we thus get that for all k such that E k n ⊂ J ε/2 and in particular Injecting this last estimate in Equation (11) and making the sum over 0 ≤ k ≤ n − 1, we get Jε Lemma 3.8. Uniformly in n, and for all 0 < η < 1, we have Proof. Applying Hölder inequality with p = 1 + η and q = 1 + 1/η and using Lemma 3.2, we have Combining the estimate (4) and Lemmas 3.7 and 3.8, we conclude that for all ε > 0 and n large enough such that nε >> 1 then

Near the atoms and conclusion
We are left to estimate the number of real zeros of fn in the neighborhood of the atoms ±α of the spectral measure µ. If ε = εn is of the form εn = n −β with 0 < β < 1/2, Proposition 3.3.1 of [Pir19] indeed show that Therefore, we can conclude that, as soon as εn is chosen of the form n −β for 0 < β < 1/5, we have which finishes the proof of Theorem 1.1. Then Corollary 1.1 follows because uniformly in x ∈ S\{0}, if nα mod π = x, then 1 − | cos(nα)| = 1 − | cos(x)| is bounded away from zero. In the last case where α / ∈ πQ, Corollary 1.2 follows from Theorem 1.1 and the regularity of α established in Lemma 3.3.

Asymptotics for a mixed spectral measure
We suppose in this section that the spectral measure µ as defined above can be written as the convex combination of a density measure and an atomic measure, i.e.
for some η ∈ [0, 1), with α k ≥ 0 for each k = 1, . . . , N . We assume that µ d admits a density ϕ w.r.t. the Lebesgue measure on [0, 2π] which satisfies the conditions : Note that the latter framework generalizes the ones of [Sam78] and [ADP19]. Indeed, taking N = 1, α = 0 and ϕ := 1 2π 1 [0,2π] corresponds to the constant correlation function ρ(·) = η ∈ (0, 1) of [Sam78]. The case η = 0 corresponds to the result obtained in [ADP19], while η = 1 corresponds to the main results of this article stated in Section 1.2. The last Theorem 4.1 shows that the contribution of the density part prevails over the one of the atomic part, and that the limit of n −1 E[N (fn, [0, 2π])] is the same as the one obtained for independent or weakly-dependent stationary Gaussian processes.
Note that the condition η ∈ [0, 1) and the non-negativity of the kernels associated with assumption A.2 ensure the non-degeneracy of fn on [0, 2π] and thus well posed nature of the last integral in Kac-Rice formula. The proof of Theorem 4.1 results from the combination of the three following lemmas. First, adapting the proof of Lemma 4 of [ADP19] and using the fact that Kn 1 = Ln 1 = 1 for all n ≥ 1, we get that in the neighborhood of zero (resp. 2π), the mean number of zeros is negligible. Finally, we show that far enough from 0, 2π and ±α, we have desired contribution. Proof. Using the explicit expressions of Kn and Ln, we have, for all t ∈ I: |Kn(t ± α)| ≤ C nε 2 n , |Ln(t ± α)| ≤ C 1 nε 2 n and 1 n K n (t ± α) ≤ C 1 nε 2 n .
The same method as in Lemma 2 of [ADP19] then yields 1 n K n * ϕ(t) ≤ C n 2 ε 3 n ( ϕ 1 + sup t∈I |ϕ(t)|) ≤ C n 2 ε 3 n , since ϕ is assumed to be uniformly bounded by A.3. Choosing εn = n −β with β ∈ (0, 1/2), the end of the proof follows the steps of the proof of Lemma 3 in [ADP19], which uses crucially the condition A.1 to have uniform convergence of the convolutions. Thus, after normalization and the cancellation of the factor 1 − η in the ratios, we get the convergence of the integrand in Kac-Rice formula to the desired universal constant, hence the result.