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Spectral Criteria for the Asymptotics of Local Functionals of Gaussian Fields and Their Application to Nodal Volumes and Critical Points

Louis Gass

TL;DR

The paper develops a spectral framework to study local functionals of vector-valued Gaussian fields by expressing variance through spectral measures and Wiener chaos expansions, enabling a general positivity criterion for the limiting variance. It then applies this approach to nodal volumes and critical points, including Euclidean random waves, deriving central limit theorems and precise fourth-chaos asymptotics. A key finding is the second-chaos cancellation phenomenon for random waves, which is explained spectrally, and the results are extended to a broad class of local functionals beyond isotropic settings. Together, the work provides a unified method for variance analysis and CLTs for geometric functionals of Gaussian fields, with broad applicability to random wave models and beyond.

Abstract

We establish a general criterion for the positivity of the variance of a chaotic component of local functionals of stationary vector-valued Gaussian fields. This criterion is formulated in terms of the spectral properties of the covariance function, without requiring integrability or isotropy. It offers a simple and robust framework for analyzing variance asymptotics in such models. We apply this approach to the study of the nodal volume and the number of critical points of a Gaussian field, proving the positivity of the limiting variance under mild conditions on the covariance function. Additionally, we examine the asymptotics of nodal volume and critical points of Euclidean random waves, deriving the central limit theorem through an analysis of the second and fourth chaotic components. As a byproduct, we unify and generalize many existing results on the volume of intersections of random waves and their critical points, bypassing the need for traditional, intricate variance computations. Our findings shed new light on the second-chaos cancellation phenomenon from a spectral perspective and can be extended to any local, possibly singular, functional of Gaussian fields.

Spectral Criteria for the Asymptotics of Local Functionals of Gaussian Fields and Their Application to Nodal Volumes and Critical Points

TL;DR

The paper develops a spectral framework to study local functionals of vector-valued Gaussian fields by expressing variance through spectral measures and Wiener chaos expansions, enabling a general positivity criterion for the limiting variance. It then applies this approach to nodal volumes and critical points, including Euclidean random waves, deriving central limit theorems and precise fourth-chaos asymptotics. A key finding is the second-chaos cancellation phenomenon for random waves, which is explained spectrally, and the results are extended to a broad class of local functionals beyond isotropic settings. Together, the work provides a unified method for variance analysis and CLTs for geometric functionals of Gaussian fields, with broad applicability to random wave models and beyond.

Abstract

We establish a general criterion for the positivity of the variance of a chaotic component of local functionals of stationary vector-valued Gaussian fields. This criterion is formulated in terms of the spectral properties of the covariance function, without requiring integrability or isotropy. It offers a simple and robust framework for analyzing variance asymptotics in such models. We apply this approach to the study of the nodal volume and the number of critical points of a Gaussian field, proving the positivity of the limiting variance under mild conditions on the covariance function. Additionally, we examine the asymptotics of nodal volume and critical points of Euclidean random waves, deriving the central limit theorem through an analysis of the second and fourth chaotic components. As a byproduct, we unify and generalize many existing results on the volume of intersections of random waves and their critical points, bypassing the need for traditional, intricate variance computations. Our findings shed new light on the second-chaos cancellation phenomenon from a spectral perspective and can be extended to any local, possibly singular, functional of Gaussian fields.
Paper Structure (32 sections, 37 theorems, 202 equations)

This paper contains 32 sections, 37 theorems, 202 equations.

Table of Contents

  1. Introduction and main results
  2. A general framework
  3. A general spectral expression for the variance
  4. A general cancellation theorem for the variance
  5. Nodal volume and critical points of Gaussian fields
  6. First chaos
  7. Second chaos
  8. Random waves and second chaos cancellation
  9. Asymptotics of the fourth chaos
  10. Hermite polynomials and Wiener chaos expansion
  11. Hermite polynomials
  12. Wiener chaos expansion
  13. Hermite expansion of distributions
  14. Positive definite functions, Hermitian form and convolution
  15. Positive Hermitian forms
  16. ...and 17 more sections

Key Result

Theorem 1.1

One has and

Theorems & Definitions (66)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 56 more