Three quantitative central limit theorems for the unbounded excursion component of a Gaussian field

For a class of smooth stationary Gaussian fields, we study the fluctuations of the volume, surface area and Euler characteristic of the unbounded excursion component in growing boxes Λₙ=[-n,n]ᵈ. We establish nondegenerate central limit theorems for all three functionals throughout the supercritical regime in every dimension d≥ 2. We further quantify these central limit theorems by establishing Berry--Esseen bounds of order O(|Λₙ|-(1-1/s)) in Kolmogorov distance, where s∈(1,2] depend on the regularity and decay assumptions. Especially for the Bargmann--Fock field, we obtain a Berry--Esseen bound of O(|Λₙ|-1/2) for all three functionals. A matching lower bound shows that this rate is sharp for the Euler characteristic. For the volume, we improve the previously known rate of O(|Λₙ|-1/2(log|Λₙ|)²ᵈ). We mainly use Stein's method and moment bounds for differences arising from white-noise resampling to finish our proof.