On the number of excursion sets of planar Gaussian fields

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.