[Paper Review] Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions

Fëdor Nazarov, Mikhail Sodin|arXiv (Cornell University)|Jul 8, 2015

Geometry and complex manifoldsMathematics20 references21 citations

TL;DR

This paper establishes almost sure asymptotic laws for the spatial distribution and number of connected components of zero sets of smooth Gaussian random functions in high dimensions. It proves that, under mild spectral conditions, the number of nodal components per unit volume converges almost surely to a deterministic constant, with explicit criteria for positivity of this constant based on the spectral measure's support and Fourier analysis.

ABSTRACT

We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains.

Motivation & Objective

• To derive almost sure asymptotic laws for the number and spatial distribution of connected components in the zero sets of smooth Gaussian random functions.
• To identify conditions on the spectral measure of translation-invariant Gaussian fields under which the nodal component density is positive and deterministic.
• To extend the theory to parametric ensembles on smooth manifolds without boundary, via a scaling and duality argument.
• To provide a criterion for the positivity of the limiting nodal density using Fourier analysis and support conditions on the spectral measure.
• To clarify the role of ergodicity and non-degeneracy in ensuring the limit is non-random and well-defined.

Proposed method

• Uses translation-invariant Gaussian processes on $\mathbb{R}^m$ with spectral measure $\rho$, ensuring $F \in C^{1+\alpha}$ via $\int |\lambda|^4 d\rho(\lambda) < \infty$.
• Applies Bulinskaya's lemma in a quantitative form to control the probability of critical points near the zero set.
• Employs the Fomin-Grenander-Maruyama theorem in the multidimensional setting to relate the nodal component count to the distribution of the gradient.
• Introduces a double scaling limit to recover the intensity function $\bar{\nu}$ from the distribution of the gradient and Hessian.
• Uses duality and analytic continuation arguments to relax the support condition in condition $(\rho 4)$ to the real-analytic closure of $\operatorname{spt}(\rho)$.
• Applies a barrier construction via cosine sums to verify condition $(\rho 4)$ when the origin lies in the interior of the convex hull of $\operatorname{spt}(\rho)$.

Experimental results

Research questions

• RQ1 Under what conditions does the number of connected components of the zero set of a smooth Gaussian function grow linearly with volume in large domains?
• RQ2 What spectral conditions on the covariance kernel ensure that the limiting nodal density is positive?
• RQ3 How can the asymptotic nodal density be recovered from the distribution of the gradient and Hessian of the Gaussian field?
• RQ4 In what cases can the support of the spectral measure be replaced by its real-analytic closure in the positivity criterion for the nodal density?
• RQ5 When does the zero set of a Gaussian field contain at least one bounded connected component, and how is this related to the spectral measure?

Key findings

• For translation-invariant Gaussian fields on $\mathbb{R}^m$, the number of connected components of the zero set in a scaled convex domain $S(R)$ grows linearly with volume, and the density converges almost surely to a deterministic constant $\nu$.
• The constant $\nu > 0$ if and only if condition $(\rho 4)$ holds, which is satisfied when the spectral measure supports a compactly supported Hermitian measure whose Fourier transform is negative on a boundary and positive inside a domain.
• Condition $(\rho 4)$ is satisfied if $0 \in \operatorname{spt}(\rho)$ or if the origin lies in the interior of the convex hull of $\operatorname{spt}(\rho)$, via a barrier construction using cosine sums.
• The spectral measure's support being contained in a quadratic hypersurface implies $(\rho 4)$ fails, so such measures yield zero nodal density.
• The real-analytic closure of $\operatorname{spt}(\rho)$ can replace the support in condition $(\rho 4)$, allowing broader applicability, e.g., when $\operatorname{spt}(\rho)$ is dense on a sphere.
• The proof shows that the nodal component density is non-random due to ergodicity of the translation action, which follows from the absence of atoms in $\rho$.