Electronic Journal of Probability

Phase singularities in complex arithmetic random waves

Federico Dalmao, Ivan Nourdin, Giovanni Peccati, and Maurizia Rossi

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Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-Itô 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. As a by-product of our analysis, we also deduce explicit bounds in smooth distances for the second order non-central results evoked above.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 71, 45 pp.

Received: 26 November 2018
Accepted: 15 May 2019
First available in Project Euclid: 28 June 2019

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Primary: 60G60: Random fields 60B10: Convergence of probability measures 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

Berry’s cancellation complex arithmetic random waves high-energy limit limit theorems Laplacian nodal intersections phase singularities Wiener chaos

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Dalmao, Federico; Nourdin, Ivan; Peccati, Giovanni; Rossi, Maurizia. Phase singularities in complex arithmetic random waves. Electron. J. Probab. 24 (2019), paper no. 71, 45 pp. doi:10.1214/19-EJP321. https://projecteuclid.org/euclid.ejp/1561687604

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