Gaussian Random Measures Generated by Berry’s Nodal Sets

Abstract

We consider vectors of random variables, obtained by restricting the length of the nodal set of Berry's random wave model to a finite collection of (possibly overlapping) smooth compact subsets of R 2 . Our main result shows that, as the energy diverges to infinity and after an adequate normalisation, these random elements converge in distribution to a Gaussian vector, whose covariance structure reproduces that of a homogeneous independently scattered random measure. A by-product of our analysis is that, when restricted to rectangles, the dominant chaotic projection of the nodal length field weakly converges to a standard Wiener sheet, in the Banach space of real-valued continuous mappings over a fixed compact set. An analogous study is performed for complex-valued random waves, in which case the nodal set is a locally finite collection of random points.

A Note on the Reduction Principle for the Nodal Length of Planar Random Waves

“…Let us now get into the notation of [NPR19,PV20]. From now on M = R 2 and we let ∆ be the Laplace operator on R 2 .…”

“…In [NPR19] and [PV20], the authors proved central limit theorems, as k → ∞, for the nodal length of planar Laplacian eigenfunctions, i.e. when M = R 2 for L f k 1 (E 0 (f k , D)), in a fixed convex body D ⊂ R 2 , using the so-called fourth moment theorem of [PT05].…”

Non-Universal Fluctuations of the Empirical Measure for Isotropic Stationary Fields on $\mathbb{S}^2 \times \mathbb{R}$

“…Several results have been given concerning the asymptotic variance, the limiting distribution and the correlation for different values of the thresholding parameter u ∈ R of Lipschitz-Killing curvatures of their excursion sets, see e.g. [9,8,22,23,25,36,34] and the references therein; see also [7,15,21,28,30] for related results on the standard flat torus and on the Euclidean plane. Some of these results entail rather surprising issues, for instance the cancellation of the leading variance terms for specific threshold values and the possibility to express wide classes of functionals as simple polynomial integrals on S 2 of the underlying fields, up to lower order terms.…”

Malliavin–Stein method: a survey of some recent developments

“…Many of hese developments largely exceed the scope of the present survey, and we invite the interested reader to consult the following general references (i)-(iii) for a more detailed presentation: (i) the webpage [1] is a constantly updated resource, listing all existing papers written around the Malliavin-Stein method; (ii) the monograph [66], written in 2012, contains a self-contained presentation of Malliavin calculus and Stein's method, as applied to functionals of general Gaussian fields, with specific emphasis on random variables belonging to a fixed Wiener chaos; (iii) the text [81] is a collection of surveys, containing an in-depth presentation of variational techniques on the Poisson space (including the Malliavin-Stein method), together with their application to asymptotic problems arising in stochastic geometry. The following more specific references (a)-(c) point to some recent developments that we find particularly exciting and ripe for further developments: (a) the papers [58,59,68,82,85,88,94] provide a representative overview of applications of Malliavin-Stein techniques to the study of nodal sets associated with Gaussian random fields on two-dimensional manifolds; (b) the papers [62,74] -and many of the reference therein -display a pervasive use of Malliavin-Stein techniques to determine rates of convergence in total variation in the Breuer-Major Theorem; (c) references [19,61] deal with the problem of tightness and functional convergence in the Breuer-Major theorem evoked at Point (b).…”