Abstract

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 ≤ ℓ } \{f \le \ell \} of additive planar Gaussian fields are bounded almost surely at the critical level ℓ c = 0 \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 ≥ 3 d \ge 3 the excursion sets have unbounded components at all levels.

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