Scars in random waves and the FGF 1/2 universality class

Abstract

We study the large-domain asymptotics of geometric observables in Berry's random wave model on $\mathbb{R}^d$. We show that, in sharp contrast with the behavior of stationary random fields with absolutely continuous spectral measures, any observable whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection belongs to a common universality class governed by a fractional Gaussian field with Hurst index $H=(1-d)/2$. This class also includes the classical stationary Poisson line process in $\mathbb{R}^d$. Our findings show that suitable raw observables of Berry's random wave (such as critical point counts or non-nodal level set volumes) have large-domain fluctuations that become arbitrarily close -- in the sense of random tempered distributions -- to those generated by a (possibly noisy) Poisson line process. This probabilistic approximation provides evidence that the large-scale filamentary patterns observed in numerical simulations of random waves -- often referred to as "scars" or "scarlets" following the numerical investigations of Heller, O'Connor and Gehlen (1987)-- may admit a natural probabilistic interpretation. In the second part of our work, we characterize the scaling limit -- in a distributional sense -- of suitable quadratic transformations of the Radon--Fourier coefficients associated with a large class of stationary fields. We show that random waves are characterized by the property that such a scaling limit is a generalized random field obtained by composing white noise on the affine Grassmannian of lines with a dimension-dependent deterministic operator. As an application of our main results, we derive explicit conditions ensuring that quadratic functionals of pullback monochromatic waves on compact Riemannian manifolds exhibit distributional limits in the fractional Gaussian universality class described above.