On the number of excursion sets of planar Gaussian fields

TL;DR

This paper generalizes the Nazarov-Sodin constant to a functional describing the number of level set components of planar Gaussian fields, using Morse theory to analyze its properties and bounds.

Contribution

It introduces a new functional for level set components, expresses it via critical point densities, and establishes its continuity and bimodality for isotropic fields.

Findings

Functional is absolutely continuous as level varies

Functional is at least bimodal for certain isotropic fields

Provides bounds for the number of level set components

Abstract

The Nazarov-Sodin constant describes the average number of nodal set components of Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.