Fluctuations of the Nodal Number in the Two-Energy Planar Berry Random Wave Model

Abstract: We investigate the fluctuations of the nodal number (count of the phase singularities) in a natural extension of the well-known complex planar Berry Random Wave Model - Berry (2002) - obtained by considering two independent real Berry Random Waves, with distinct energies $E_1, E_2 \to \infty$ (at possibly $\neq$ speeds). Our framework relaxes the conditions used in Nourdin, Peccati and Rossi (2019) where the energies were assumed to be identical ($E_1 \equiv E_2$). We establish the asymptotic equivalence of the nodal number with its 4-th chaotic projection and prove quantitative Central Limit Theorems (CLTs) in the 1-Wasserstein distance for the univariate and multivariate scenarios. We provide a corresponding qualitative theorem on the convergence to the White Noise in a sense of random distributions. We compute the exact formula for the asymptotic variance of the nodal number with exact constants depending on the choice of the subsequence. We provide a simple and complete characterisation of this dependency through introduction of the three asymptotic parameters: $r^{log}$, $r$, $r^{exp}$. The corresponding claims in the one-energy model were established in Nourdin, Peccati and Rossi (2019), Peccati and Vidotto (2020), Notarnicola, Peccati and Vidotto (2023), and we recover them as a special case of our results. Moreover, we establish full-correlations with polyspectra, which are analogues of the full-correlation with tri-spectrum that was previously observed for the nodal length in Vidotto (2021).