Phase Singularities in Complex Arithmetic Random Waves

TL;DR

This paper characterizes the second order fluctuations of phase singularities in complex arithmetic random waves on a torus, revealing non-universal behaviors influenced by spectral measure arithmetic properties.

Contribution

It provides a complete second order high-energy analysis of phase singularities, extending previous real-valued results to the complex setting with new fluctuation insights.

Findings

Universal law of large numbers for phase singularities.

Non-universal, non-central fluctuations influenced by spectral measure arithmetic.

Asymptotic variance characterized via Kac-Rice kernel and combinatorial methods.

Abstract

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-It\^o chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations.