We prove a Central Limit Theorem for the number of zeros of random trigonometric polynomials of the form K −1/2 K n=1 a n cos(nt), being (a n ) n independent standard Gaussian random variables. In particular we show that the variance is equivalent to V 2 Kπ,Our approach is based on the Hermite/Wiener-Chaos decomposition for square-integrable functionals of a Gaussian process and on Rice Formula for zero counting.Nous montrons un Théorème de la Limite Central pour le nombre de racines d'un polynôme trigonométrique aléatoire de la forme K −1/2 K n=1 a n cos(nt), ici les a n sont des variables aléatoires Gaussiennes standard et indépendantes. En particulier, nos démontrons que la variance asymptotique du nombre de racines estéquivalentà V 2 Kπ, pour une certaine constante V > 0, lorsque K → ∞. Ce dernier résultat aété récemment démontré par Su & Shao dans [23]. Notre approche utilise la décomposition dans le chaos d'Itô-Wiener d'une fonctionnelle non linéaire de carré intégrable et la formule de Rice.

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-Itô chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations.

In this note, we find the asymptotic main term of the variance of the number of roots of Kostlan-Shub-Smale random polynomials and prove a central limit theorem for this number of roots as the degree goes to infinity. r é s u m éDans cette note, nous calculons un equivalent de la variance asymptotique du nombre de racines réelles des polynômes aléatoires de Kostlan-Shub-Smale et nous démontrons un théorème de la limite centrale pour ce même nombre quand le degré tend vers l'infini.

Consider a flock of birds that fly interacting between them. The interactions are modelled through a hierarchical system in which each bird, at each time step, adjusts its own velocity according to his past velocity and a weighted mean of the relative velocities of its superiors in the hierarchy. We consider the additional fact, that each of the birds can fail to see any of its superiors with certain probability, that can depend on the distances between them. For this model with random interactions we prove that the flocking phenomena, obtained for similar deterministic models, holds true.

We consider random trigonometric polynomials of the form f n (t) := 1≤k≤n a k cos(kt) + b k sin(kt), whose entries (a k ) k≥1 and (b k ) k≥1 are given by two independent stationary Gaussian processes with the same correlation function ρ. Under mild assumptions on the spectral function ψ ρ associated with ρ, we prove that the expectation of the number N n ([0, 2π]) of real roots of f n in the interval [0, 2π] satisfiesThe latter result not only covers the well-known situation of independent coefficients but allow us to deal with long range correlations. In particular it englobes the case where the random coefficients are given by a fractional Brownian noise with any Hurst parameter.