Communications in Number Theory and Physics

Volume 16 (2022)

Number 1

Intermediate and small scale limiting theorems for random fields

Pages: 1 – 34



Dmitry Beliaev (Mathematical Institute, University of Oxford, United Kingdom)

Riccardo W. Maffucci (École Polytechnique Fédérale de Lausanne (EPFL), Switzerland)


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.


Gaussian fields, random waves, nodal lines, coupling, persistence, large deviations

2010 Mathematics Subject Classification

60F10, 60G15, 60G60

Both authors were partially supported by the Engineering & Physical Sciences Research Council (EPSRC) Fellowship EP/M002896/1.

Dmitry Beliaev was partially supported by the Ministry of Science and Higher Education of Russia, grant 075-15-2019-1620.

Riccardo W. Maffucci was supported by Swiss National Science Foundation project 200021_184927.

Received 8 July 2019

Accepted 15 August 2021

Published 1 February 2022