Boundedness of the nodal domains of additive Gaussian fields

TL;DR

This paper proves that for a class of additive Gaussian fields in two dimensions, the excursion sets are almost surely bounded at the critical level, providing new examples of non-positively correlated fields with this property.

Contribution

It establishes the boundedness of nodal domains at the critical level for additive Gaussian fields without assuming positive correlations, filling a gap in the understanding of Gaussian field topology.

Findings

Excursion sets are bounded at the critical level in 2D additive Gaussian fields.

Provides first examples of non-positively correlated fields with bounded nodal domains.

In dimensions 3 and higher, excursion sets have unbounded components at all levels.

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 $\{f \le \ell\}$ 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.