On the number of excursion sets of planar Gaussian fields
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On the number of excursion sets of planar Gaussian fields Dmitry Beliaev1
· Michael McAuley1
· Stephen Muirhead2,3
Received: 25 October 2018 © The Author(s) 2020
Abstract 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. Keywords Gaussian fields · Nodal set · Level sets · Critical points Mathematics Subject Classification 60G60 · 60G15 · 58K05
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”.
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Michael McAuley [email protected] Dmitry Beliaev [email protected] Stephen Muirhead [email protected]
1
Mathematical Institute, University of Oxford, Oxford, UK
2
Department of Mathematics, King’s College London, London, UK
3
Present Address: School of Mathematical Sciences, Queen Mary University of London, London, UK
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D. Beliaev et al.
1 Introduction 1.1 The Nazarov–Sodin constant Let f : R2 → R be a continuous stationary planar Gaussian field normalised to have zero mean and unit variance. The nodal set of f is the random set N = x ∈ R2 : f (x) = 0 . Let κ : R2 → [−1, 1] denote the covariance kernel of f , i.e. κ(x) = E[ f (x) f (0)]. We assume throughout that κ is C 4+ , which ensures that almost surely f is C 2+ . Since κ is positive definite, continuous and κ(0) = 1, by Bochner’s theorem there exists a probability measure ρ such that κ(x) =
R2
eit,x dρ(t);
(1)
this is known as the spectral measure of f , and must be Hermitian (that is, ρ(−A) = ρ(A) for all Borel sets A). Since the distribution of f is uniquely determined by its covariance function (Kolmogorov’s theorem), (1) shows that the distribution of f is uniquely determined by ρ. The geometric properties of N are of interest, in part, because in the case that f is a random eigenfunction of the Laplacian they relate to a significant conjecture in the physics literature: the Berry conjecture [4]. A summary of this conjecture and other research on this topic may be found in [17]. One of the main analytical results concerning this set, due to Nazarov and Sodin [13,15], states that the number of components of N in a large domain scales like the area of the domain. Specifically, if N R denotes the number of components of N inside the centred ball of radius R > 0, then provided f is ergodic, there exists a constant c L S = c L S (ρ) ≥ 0 such that N R /(π R 2 )
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