On the nodal structures of random fields -- a decade of results

Wigman, Igor

doi:10.48550/arxiv.2206.10020

Cited by 3 publications

(5 citation statements)

References 65 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For all these results and a global overview of the subject we refer the reader to [17] and the references therein. A stream of results has also emerged in the past twenty years regarding nodal sets of random waves (i.e., random linear combinations of eigenfunctions); for this we refer the reader to the recent survey [25] and the references therein.…”

Section: Eigenfunctions and Nodal Setsmentioning

confidence: 99%

Nodal sets of eigenfunctions of sub-Laplacians

Eswarathasan¹,

Letrouit²

2023

Preprint

View full text Add to dashboard Cite

Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In this note, we initiate the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians. A standard example is the sum of squares of bracket-generating vector fields on compact quotients of the Heisenberg group. Our results show that nodal sets behave in an anisotropic way which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation and desingularization at singular points. Furthermore, we provide a simple example demonstrating that for sub-Laplacians, the Hausdorff measure of nodal sets of eigenfunctions cannot be bounded above by √ λ, which is the bound conjectured by Yau for Laplace-Beltrami operators on smooth manifolds.

show abstract

Section: Eigenfunctions and Nodal Setsmentioning

confidence: 99%

Nodal sets of eigenfunctions of sub-Laplacians

Eswarathasan¹,

Letrouit²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Wiener-Itô chaoses. The result in Theorem 2.1 can be interpreted in terms of the L 2 (Ω) expansion of the nodal length into Wiener-Itô chaoses (see also the discussion in [34]), which are orthogonal spaces spanned by Hermite polynomials. First of all, we recall that the Hermite polynomials H q (x) are defined by H 0 (x) = 1, and for q = 2, 3, .…”

Section: Interpretation In Terms Ofmentioning

confidence: 99%

“…For example, as it has been shown in [27], it should be a good model for f −1 λ (0) ∩ B r λ , in particular for the nodal length lying inside a shrinking geodesic ball B r λ of radius slightly above the Planck scale: r λ ≈ C √ λ with C ≫ 0 sufficiently big. For a recent survey on nodal structures of random fields see [34].…”

Section: Introductionmentioning

confidence: 99%

No smooth Phase Transition for the Nodal Length of Band-limited Spherical Random Fields

Todino¹

2023

Preprint

View full text Add to dashboard Cite

In this paper, we investigate the variance of the nodal length for band-limited spherical random waves. When the frequency window includes a number of eigenfunctions that grows linearly, the variance of the nodal length is linear with respect to the frequency, while it is logarithmic when a single eigenfunction is considered. Then, it is natural to conjecture that there exists a smooth transition with respect to the number of eigenfunctions in the frequency window; however, we show here that the asymptotic variance is logarithmic whenever this number grows sublinearly, so that the window "shrinks". The result is achieved by exploiting the Christoffel-Darboux formula to establish the covariance function of the field and its first and second derivatives. This allows us to compute the two-point correlation function at high frequency and then to derive the asymptotic behaviour of the variance.

show abstract

“…The analysis of level curves for random fields is a very classical topic in stochastic geometry. In particular, many efforts have focussed on the investigation of level-zero curves (i.e., nodal lines) in the case of random eigenfunctions, in the high-frequency regime where eigenvalues are assumed to diverge to infinity; see for instance [Wig10,MPRW16,NPR19], or more generally [Wig22] for a recent overview. In the same high-energy regime, other functionals for random eigenfunctions (including excursion area, the Euler-Poincaré characteristic, the number of critical points) have also been widely investigated, see for instance [Mar2022]; on the other hand, these same functionals have also been considered by different authors in the asymptotic regime where the spatial domain of the field is assumed as growing, notable examples being [KL01] (for level curves) and [EL2016] (for the Euler-Poincaré characteristic).…”

Section: Background and Notationmentioning

confidence: 99%

“…The literature on the geometry of random fields on manifolds has become vast over the last decade, see for instance [Ros19], [Mar2022], [Wig22] for some recent surveys. Much of the literature has concentrated on the high-frequency geometry for random eigenfunctions, in the case of random fields on the sphere (or on other Riemannian manifolds, for instance the torus) with no temporal dependence.…”

Section: A Comparison With the High-energy Regime Literaturementioning

confidence: 99%

Fluctuations of Level Curves for Time-Dependent Spherical Random Fields

Marinucci¹,

Rossi²,

Vidotto³

2022

Preprint

View full text Add to dashboard Cite

The investigation of the behaviour for geometric functionals of random fields on manifolds has drawn recently considerable attention. In this paper, we extend this framework by considering fluctuations over time for the level curves of general isotropic Gaussian spherical random fields. We focus on both long and short memory assumptions; in the former case, we show that the fluctuations of u-level curves are dominated by a single component, corresponding to a second-order chaos evaluated on a subset of the multipole components for the random field. We prove the existence of cancellation points where the variance is asymptotically of smaller order; these points do not include the nodal case u = 0, in marked contrast with recent results on the high-frequency behaviour of nodal lines for random eigenfunctions with no temporal dependence. In the short memory case, we show that all chaoses contribute in the limit, no cancellation occurs and a Central Limit Theorem can be established by Fourth-Moment Theorems and a Breuer-Major argument.

show abstract

On the nodal structures of random fields -- a decade of results

Cited by 3 publications

References 65 publications

Nodal sets of eigenfunctions of sub-Laplacians

Nodal sets of eigenfunctions of sub-Laplacians

No smooth Phase Transition for the Nodal Length of Band-limited Spherical Random Fields

Fluctuations of Level Curves for Time-Dependent Spherical Random Fields

Contact Info

Product

Resources

About