Nodal Statistics of Planar Random Waves

TL;DR

This paper proves that the nodal length and intersection counts of Berry's random planar waves follow a Gaussian distribution in the high-energy limit, contrasting with non-Gaussian behaviors in other models, and confirms variance asymptotics for these statistics.

Contribution

It establishes a Central Limit Theorem for nodal length and intersection counts of Berry's model, extending the understanding of high-energy eigenfunctions in planar domains.

Findings

Nodal length and intersection counts are asymptotically Gaussian.

Variance asymptotics for nodal statistics are rigorously confirmed.

Results connect Berry's model with general Gaussian wave behaviors.

Abstract

We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0$, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ($E\to \infty$). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat $2$-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by Canzani and Hanin (2016), in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002).