Discretisation schemes for level sets of planar Gaussian fields

Dmitry Beliaev, Stephen Muirhead

Published 2017-02-07Version 1

We study four discretisation schemes that extract information about level sets of planar Gaussian fields. Each scheme recovers information up to 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; these give complete topological information about the level sets on either a local or global scale. As an application, we improve recent results 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 an approximation of the Nazarov-Sodin constant for planar Gaussian fields.