  For a smooth, stationary Gaussian field $f$ on Euclidean space with fast
correlation decay, there is a critical level $\ell_c$ such that the excursion
set $\{f\geq\ell\}$ contains a (unique) unbounded component if and only if
$\ell<\ell_c$. 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
$\ell<\ell_c$). In higher dimensions the results hold at all sufficiently low
levels (all $\ell<-\ell_c<\ell_c$) 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.
