Boundedness of the nodal domains of additive Gaussian fields

We study the connectivity of the excursion sets of additive Gaussian fields, i.e.\ stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets $\{f \le \ell\}$ of additive planar Gaussian fields are bounded almost surely at the critical level $\ell_c = 0$. Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension $d \ge 3$ the excursion sets have unbounded components at all levels.