  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.
