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

For a smooth, stationary Gaussian field f on Euclidean space with fast correlation decay, there is a critical level ℓc such that the excursion set {f≥ℓ} contains a (unique) unbounded component if and only if ℓ<ℓc. We prove central limit theorems for the volume, surface area and Euler characteristic of this unbounded component restricted to a growing box. For planar fields, the results hold at all supercritical levels (i.e., all ℓ<ℓc). In higher dimensions the results hold at all sufficiently low levels (all ℓ<−ℓc<ℓc) but could be extended to all supercritical levels by proving the decay of truncated connection probabilities. Our proof is based on the martingale central limit theorem.