Small Scale CLTs for the Nodal Length of Monochromatic Waves

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.