On the number of excursion sets of planar Gaussian fields

Dmitry Beliaev, Michael McAuley, Stephen Muirhead

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)655-698
Number of pages44
JournalProbability Theory and Related Fields
Issue number3-4
Publication statusPublished - 1 Dec 2020
Externally publishedYes


  • Critical points
  • Gaussian fields
  • Level sets
  • Nodal set


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