Boundedness of the nodal domains of additive Gaussian fields

TL;DR

This work proves that for planar additive Gaussian fields with mild smoothness and correlation decay, the critical level for percolation is $\ell_c=0$, and the nodal domains of $\{f\le\ell\}$ are almost surely bounded when $\ell\le0$ (with a unique unbounded component for $\ell>0$). It handles non-positively correlated fields, providing the first such planar examples, and contrasts with the $d\ge3$ case where unbounded components appear at all levels. The approach reduces the problem to extreme-value theory for one-dimensional Gaussian processes, employing scaling limits for $\sup g$, concentration bounds, and point-process convergence of extrema, plus ergodicity to obtain blocking arguments. The analysis also clarifies the critical phase, showing box-crossing behavior depends on $K_1(0)$ vs $K_2(0)$ and establishing a $1/\sqrt{\log R}$ scale for the critical window. Overall, the results bridge Gaussian-field percolation, extreme-value theory, and ergodic arguments to characterize planar nodal geometry in additive fields.

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.

Figures (1)

• Figure 1: The nodal domains $\{f \le 0\}$ (in black) for the additive Gaussian fields with $K_1(x) = K_2(x) = e^{-x^2}$ (left frame) and $K_1(x) = K_2(x) = \cos(x) e^{-x^2}$ (right frame). Theorem \ref{['t:main']} states that these sets have bounded connected components almost surely. Note that the first field is positively-correlated while the second field is not.

Theorems & Definitions (21)

• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Proposition 2.1: Scaling limit of the supremum; llr83
• Proposition 2.2: Concentration of the supremum; see tan15