Local universality for real roots of random trigonometric polynomials

Abstract

We consider random trigonometric polynomials of the form f n (x, y) = 1≤k,l≤n a k,l cos(kx) cos(ly), where the entries (a k,l) k,l≥1 are i.i.d. random variables that are centered with unit variance. We investigate the length K (f n) of the nodal set Z K (f n) of the zeros of f n that belong to a compact set K ⊂ R 2. We first establish a local universality result, namely we prove that, as n goes to infinity, the sequence of random variables n K/n (f n) converges in distribution to a universal limit which does not depend on the particular law of the entries. We then show that at a macroscopic scale, the expectation of [0,π] 2 (f n)/n also converges to an universal limit. Our approach provides two main byproducts: (i) a general result regarding the continuity of the volume of the nodal sets with respect to C 1-convergence which refines previous findings of [RS01, IK16, ADL + 15] and (ii) a new strategy for proving small ball estimates in random trigonometric models, providing in turn uniform local controls of the nodal volumes.

Expected number of real zeroes of random Taylor series

“…In fact, by Theorem 3.1, both (19) and (20) hold uniformly over z ∈ Q R , where in the latter case we have to employ (18). Taking the quotient of (19) and (20), recalling that γ > 0 and using (17) we arrive at the required estimate |v(1 − a n z)| cv(|1 − a n z|).…”

Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature

“…Indeed, we have P n (0) = f n,0 ξ 0 , so there can be no central limit theorem if ξ 0 is not Gaussian. For other random analytic functions, functional limit theorems of the above type appeared in [28,3,24,16].…”

“…By Theorem 2.2, the latter process converges weakly to Z γ(t) (·) on the space A real (D R ), as n → ∞. We may take R > δ, so that the interval [0, δ] is contained in the interior of the disk D R of radius R. To pass to the distributional convergence of real zeroes, we employ the continuous mapping theorem in the same way as it was done in [24]. By Lemma 4.1 therein, the map which assigns to a function f ∈ A real (D R )\{0} the number of zeroes of f in the interval [0, δ] is locally constant (hence, continuous) on the set of all analytic functions which do not vanish at 0, δ and have no multiple zeroes in the interval [0, δ].…”

“…This set has full measure w.r.t. the law of Z γ(t) (the a.s. absence of multiple zeroes follows from the Bulinskaya lemma; see, e.g., [24,Lemma 4.3]). Hence, the continuous mapping theorem implies the distributional convergence of N n [t, t + δ/ √ n] to the number of zeroes of Z γ(t) in [0, δ] as n → ∞.…”

Non universality for the variance of the number of real roots of random trigonometric polynomials

“…joint distribution of roots at microscopic scales) and universality of the expectation at a global scale have been achieved successfully. Concerning local universality, we refer to [18,7,13] and for expectation to [16,9,10]. Very often, the extension to the global scale of the microscopic distribution of the roots is not an easy task, and one needs first to provide suitable estimates for the so-called phenomenon of repulsion of zeros.…”