# Boundary effect on the nodal length for Arithmetic Random Waves, and spectral semi-correlations

## TL;DR

This paper investigates how boundary effects influence the nodal length of arithmetic random waves in a square billiard, revealing that spectral properties and number theory significantly affect nodal deficiencies or surpluses.

## Contribution

It introduces boundary-adapted arithmetic random waves and links the expected nodal length to spectral semi-correlations, providing a detailed asymptotic analysis and connecting spectral properties with number theory.

## Key findings

- Nodal length asymptotics depend on arithmetic properties of energy levels.
- Spectral semi-correlations are key in understanding boundary effects.
- Precise asymptotic expansion of expected nodal length derived.

## Abstract

We test M. Berry's ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions ("boundary-adapted arithmetic random waves"). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels. It is found that the precise nodal deficiency or surplus of the nodal length depends on arithmetic properties of the energy levels, in an explicit way. To obtain the said results we apply the Kac-Rice method for computing the expected nodal length of a Gaussian random field. Such an application uncovers major obstacles, e.g. the occurrence of "bad" subdomains, that, one hopes, contribute insignificantly to the nodal length. Fortunately, we were able to reduce this contribution to a number theoretic question of counting the "spectral semi-correlations", a concept joining the likes of "spectral correlations" and "spectral quasi-correlations" in having impact on the nodal length for arithmetic dynamical systems. This work rests on several breakthrough techniques of J. Bourgain, whose interest in the subject helped shaping it to high extent, and whose fundamental work on spectral correlations, joint with E. Bombieri, has had a crucial impact on the field.