Small Scale CLTs for the Nodal Length of Monochromatic Waves

TL;DR

This paper proves a Central Limit Theorem for the nodal length of Gaussian monochromatic random waves on Riemann surfaces, under certain shrinking ball conditions, extending previous models and improving variance estimates.

Contribution

It introduces a novel intrinsic Gaussian coupling bound and applies it to establish CLTs for nodal lengths in new geometric settings.

Findings

CLT holds for nodal length when radius grows slower than (log λ)^{1/25}

Improved estimates for variance of nodal length in pullback random waves

Application to phase transitions in arithmetic random waves on shrinking balls

Abstract

We consider the nodal length $L(\lambda)$ of the restriction to a ball of radius $r_\lambda$ of a {\it Gaussian pullback monochromatic random wave} of parameter $\lambda>0$ associated with a Riemann surface $(\mathcal M,g)$ without conjugate points. Our main result is that, if $r_\lambda$ grows slower than $(\log \lambda)^{1/25}$, then (as $\lambda\to \infty$) the length $L(\lambda)$ verifies a Central Limit Theorem with the same scaling as Berry's random wave model -- as established in Nourdin, Peccati and Rossi (2019). Taking advantage of some powerful extensions of an estimate by B\'erard (1986) due to Keeler (2019), our techniques are mainly based on a novel intrinsic bound on the coupling of smooth Gaussian fields, that is of independent interest, and moreover allow us to improve some estimates for the nodal length asymptotic variance of pullback random waves in Canzani and Hanin (2016). In order to demonstrate the flexibility of our approach, we also provide an application to phase transitions for the nodal length of arithmetic random waves on shrinking balls of the $2$-torus.