Let be a stationary centered Gaussian process. For any , let denote the counting measure of . Under suitable assumptions on the regularity of f and the decay of its correlation function at infinity, we derive the asymptotics as of the central moments of the linear statistics of . In particular, we derive an asymptotics of order for the p-th central moment of the number of zeros of f in . As an application, we prove a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures . More precisely, after a proper rescaling, converges almost surely towards the Lebesgue measure in weak-∗ sense. Moreover, the fluctuation of 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 . 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 , 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.
This work was supported by the French National Research Agency grants UniRaNDom (ANR-17-CE40-0008) and SpInQS (ANR-17-CE40-0011) and by the Israeli Science Foundation Grants 382/15 and 501/18.
Thomas Letendre thanks Julien Fageot for useful discussions about Fernique’s Theorem, Benoit Laslier for his help in the proof of Lemma C.2 and Hugo Vanneuville for pointing out the relation between ergodicity and decay of correlations. The authors are grateful to Damien Gayet for suggesting they write this paper in the first place, and to Misha Sodin for bringing to their attention the intrinsic interest of clustering properties for k-point functions. They also thank Jean-Yves Welschinger for his support and Louis Gass for spotting an error in an earlier version of Lemma 7.8. Finally, the authors thank the anonymous referee for their careful reading of the paper, and their comments that helped to improve the exposition of the main results.
"Zeros of smooth stationary Gaussian processes." Electron. J. Probab. 26 1 - 81, 2021. https://doi.org/10.1214/21-EJP637