Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients

الأصل · EN

Let Xₙ be a random trigonometric polynomial of degree N with iid coefficients and let Zₙ(I) denote the (random) number of its zeros lying in the compact interval I. Recently, a number of important advances were made in the understanding of the asymptotic behaviour of Zₙ(I) as N→∞, in the case of standard Gaussian coefficients. The main theorem of the present paper is a universality result, that states that the limit of Zₙ(I) does not really depend on the exact distribution of the coefficients of Xₙ. More precisely, assuming that these latter are iid with mean zero and unit variance and have a density satisfying certain conditions, we show that Zₙ(I) converges in distribution toward Z(I), the number of zeros within I of the centered stationary Gaussian process admitting the cardinal sine for covariance function.