The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration

TL;DR

This paper proves the existence, uniqueness, and volume concentration of a giant component in supercritical excursion sets of spherical Gaussian fields, revealing similarities to percolation phenomena and introducing new decoupling inequalities.

Contribution

It establishes the first rigorous results on giant components in spherical Gaussian ensembles, with quantitative bounds and novel decoupling techniques.

Findings

Existence and uniqueness of a giant component in supercritical regimes.

Quantitative bounds on volume fluctuations of the giant component.

Giant components exhibit properties similar to supercritical percolation.

Abstract

We establish the existence and uniqueness of a well-concentrated giant component in the supercritical excursion sets of three important ensembles of spherical Gaussian random fields: Kostlan's ensemble, band-limited ensembles, and the random spherical harmonics. Our main results prescribe quantitative bounds for the volume fluctuations of the giant that are essentially optimal for non-monochromatic ensembles, and suboptimal but still strong for monochromatic ensembles. Our results support the emerging picture that giant components in Gaussian random field excursion sets have similar large-scale statistical properties to giant components in supercritical Bernoulli percolation. The proofs employ novel decoupling inequalities for spherical ensembles which are of independent interest.