Vol. 3, No. 1, 2022

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Fluctuations of the number of excursion sets of planar Gaussian fields

Dmitry Beliaev, Michael McAuley and Stephen Muirhead

Vol. 3 (2022), No. 1, 105–144
Abstract

For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area R2. The mean number of components is known to be of order R2 for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann–Fock field), and for certain levels , these random variables have fluctuations of order at least R, and hence variance of order at least R2. In particular this holds for excursion sets when is in some neighbourhood of zero, and it holds for excursion/level sets when  is sufficiently large. We prove stronger fluctuation lower bounds of order Rα for α [1,2] in the case that the spectral density has a singularity at the origin. Finally we show that the number of excursion/level sets for the random plane wave at certain levels has fluctuations of order at least R32, and hence variance of order at least R3. We expect that these bounds are of the correct order, at least for generic levels.

Keywords
Gaussian fields, level sets, excursion sets, nodal sets, fluctuations, variance bounds
Mathematical Subject Classification
Primary: 58K05, 60G15, 60G60
Milestones
Received: 2 December 2020
Revised: 20 August 2021
Accepted: 15 September 2021
Published: 11 May 2022
Authors
Dmitry Beliaev
Mathematical Institute
University of Oxford
United Kingdom
Michael McAuley
Mathematical Institute
University of Oxford
United Kingdom
Department of Mathematics and Statistics
University of Helsinki
Finland
Stephen Muirhead
School of Mathematical Sciences
Queen Mary University of London
United Kingdom
School of Mathematics and Statistics
University of Melbourne
Australia