Theory of Probability and Mathematical Statistics
Boundedness of the nodal domains of additive Gaussian fields
S. Muirhead
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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} of additive planar Gaussian fields are bounded almost surely at the critical level lc=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 the excursion sets have unbounded components at all levels.
Keywords: Gaussian fields, level sets, nodal domains, percolation
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