Zeros of smooth stationary Gaussian processes

Abstract

Let $f:\mathbb{R}\to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let ${\mathrm{\nu }}_R$ denote the counting measure of $\{x\in \mathbb{R}\mid f(Rx)=0\}$. Under suitable assumptions on the regularity of f and the decay of its correlation function at infinity, we derive the asymptotics as $R\to +\mathrm{\infty }$ of the central moments of the linear statistics of ${\mathrm{\nu }}_R$. In particular, we derive an asymptotics of order $R^{\frac{p}{2}}$ for the p-th central moment of the number of zeros of f in $[0,R]$. As an application, we prove a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures ${\mathrm{\nu }}_R$. More precisely, after a proper rescaling, ${\mathrm{\nu }}_R$ converges almost surely towards the Lebesgue measure in weak-∗ sense. Moreover, the fluctuation of ${\mathrm{\nu }}_R$ around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the k-point function of the zero point process of f, for any $k\ge 2$. Our analysis yields two results of independent interest. First, we derive an equivalent of this k-point function near any point of the large diagonal in ${\mathbb{R}}^k$, thus quantifying the short-range repulsion between zeros of f. Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of f.