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.
Highlights
1.1 Overview and main resultsLet T := R2/Z2 be the two-dimensional flat torus, and define ∆ = ∂2/∂x21 + ∂2/∂x22 to be the associated Laplace-Beltrami operator
In order to accomplish this task, we will extend and generalise the approach initiated in [M-P-R-W], in particular by providing a new set of techniques that allow one to control residual terms arising in Wiener-Itô chaotic expansions, as well as to deduce explicit bounds in smooth distances for second order fluctuations
In order to understand our setting, recall that the eigenvalues of −∆ on T are the positive reals of the form En := 4π2n, where n = a2 + b2 for some a, b ∈ Z
Summary
The main result of the present work is the following exact characterization of the first and second order behaviours of In, as defined by (1.4), in the high-energy limit As discussed below, it is an extension of the results proved in [K-K-W, M-P-R-W], providing a rigorous description of the Berry’s cancellation phenomenon [Be3] in the context of phase singularities of complex random waves. Our statement (whose elementary proof is omitted) yields a complete characterization of the distribution of the vector-valued process Tn, as a two-dimensional field whose components are independent and identically distributed real arithmetic random waves, in the sense of (1.13). A(n) ∪ A(n) and A(m) ∪ A(m) are stochastically independent; this is the same as assuming that the two vector-valued fields Tn and Tm are stochastically independent
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
