Small Scale CLTs for the Nodal Length of Monochromatic Waves

We consider the nodal length L(λ ) of the restriction to a ball of radius r_λ of a Gaussian pullback monochromatic random wave of parameter λ >0 associated with a Riemann surface (ℳ,g) without conjugate points. Our main result is that, if r_λ grows slower than (logλ )^1/25 , then (as λ→∞ ) the length L(λ ) verifies a Central Limit Theorem with the same scaling as Berry’s random wave model—as established in Nourdin et al. (Commun Math Phys 369(1):99–151, 2019). Taking advantage of some powerful extensions of an estimate by Bérard (Mathematische Zeitschrift 155:249–276, 1977) due to Keeler (A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points, 2019. arXiv:1905.05136 ), 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 (Commun Math Phys 378:1677–1712, 2020). 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.