Scars in random waves and the ${\bf FGF(\frac12)}$ 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 -- under suitable non-degeneracy assumptions -- belongs to a common universality class governed by a fractional Gaussian field with Hurst index $H=\frac{1-d}{2}$. This class also includes the classical stationary Poisson line process in $\mathbb{R}^d$. This result is consistent with the description of the filamentary visual patterns -- often described as scars'' orscarlets'' -- observed in numerical simulations of random waves since the work of Heller, O'Connor and Gehlen (1987), in the precise sense that our findings show that suitable observables of Berry's random wave exhibit fluctuations that, in the large-domain limit, become arbitrarily close to those generated by a Poisson line process. It also suggests that the emergence of linear scar-like patterns in random waves is a phenomenon that persists in any dimension. In the second part of our work, we characterize the scaling limit -- in a distributional sense -- of suitable quadratic transformations of the {\it 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.