Journal article

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.

University of Melbourne Researchers


Awarded by Engineering & Physical Sciences Research Council (EPSRC) Fellowship

Awarded by EPSRC

Funding Acknowledgements

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".