On the number of excursion sets of planar Gaussian fields
D Beliaev, M McAuley, S Muirhead
Probability Theory and Related Fields | Springer | Published : 2020
The Nazarov–Sodin constant describes the average number of nodal set components of smooth 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.
Awarded by Engineering & Physical Sciences Research Council (EPSRC) Fellowship
Awarded by EPSRC
Dmitry Beliaev was supported by the Engineering & Physical Sciences Research Council (EPSRC) Fellowship EP/M002896/1. Stephen Muirhead was supported by the EPSRC Grant EP/N009436/1 "The many faces of random characteristic polynomials".