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 $Λ_n=[-n,n]^d$. We establish nondegenerate central limit theorems for all three functionals throughout the supercritical regime in every dimension $d\geq 2$. We further quantify these central limit theorems by establishing Berry--Esseen bounds of order $O\bigl(|Λ_n|^{-(1-1/s)}\bigr)$ in Kolmogorov distance, where $s\in(1,2]$ depend on the regularity and decay assumptions. Especially for the Bargmann--Fock field, we obtain a Berry--Esseen bound of $O\bigl(|Λ_n|^{-1/2}\bigr)$ 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 $O\bigl(|Λ_n|^{-1/2}(\log|Λ_n|)^{2d}\bigr)$. We mainly use Stein's method and moment bounds for differences arising from white-noise resampling to finish our proof.