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

TL;DR

This paper investigates the asymptotic behavior of the spatial distribution and the number of connected components of zero sets in smooth Gaussian random functions across various domains and ensembles, revealing fundamental statistical properties.

Contribution

It introduces new asymptotic laws describing the distribution and topology of zero sets of Gaussian functions in high-degree polynomial and translation-invariant cases.

Findings

Asymptotic laws for zero set distribution established

Quantitative descriptions of connected components provided

Results apply to polynomials on spheres, tori, and Euclidean spaces

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.