# Discretisation schemes for level sets of planar Gaussian fields

## TL;DR

This paper introduces four discretisation schemes for analyzing level sets of planar Gaussian fields, improving topological understanding and estimating excursion domains, with some schemes being novel contributions.

## Contribution

The work presents new discretisation methods for level sets of Gaussian fields, including the first schemes to provide global topological information for the random plane wave's nodal set and a scheme to estimate mean excursion domains.

## Key findings

- Enhanced topological analysis of Gaussian field level sets.
- Improved Russo-Seymour-Welsh estimates for nodal sets.
- Introduction of new discretisation schemes for Gaussian fields.

## Abstract

Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic compact manifolds. Our work is motivated by questions about the geometry of such random functions, in particular relating to the structure of their nodal and level sets. We study four discretisation schemes that extract information about level sets of planar Gaussian fields. Each scheme recovers information up to a different level of precision, and each requires a maximum mesh-size in order to be valid with high probability. The first two schemes are generalisations and enhancements of similar schemes that have appeared in the literature [BG16, MW07]; these give complete topological information about the level sets on either a local or global scale. As an application, we improve the results in [BG16] on Russo-Seymour-Welsh estimates for the nodal set of positively-correlated planar Gaussian fields. The third and fourth schemes are, to the best of our knowledge, completely new. The third scheme is specific to the nodal set of the random plane wave, and provides global topological information about the nodal set up to `visible ambiguities'. The fourth scheme gives a way to approximate the mean number of excursion domains of planar Gaussian fields.