2019/06/02 by Dmitry Beliaev, Beliaev, Dmitry, Riccardo W. Maffucci +1
Mathematics · #60F10 #60G15 #60G60 #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1906.00435
openalex publication_date 2019/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the nodal lines of random eigenfunctions of the Laplacian on the torus, the so called 'arithmetic waves'. To be more precise, we study the number of intersections of the nodal line with a straight interval in a given direction. We are interested in how this number depends on the length and direction of the interval and the distribution of spectral measure of the random wave. We analyse the second factorial moment in the short interval regime and the persistence probability in the long interval regime. We also study relations between the Cilleruelo and Cilleruelo-type fields. We give an explicit coupling between these fields which on mesoscopic scales preserves the structure of the nodal sets with probability close to one.