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2020 Smoothness and monotonicity of the excursion set density of planar Gaussian fields
Dmitry Beliaev, Michael McAuley, Stephen Muirhead
Electron. J. Probab. 25: 1-37 (2020). DOI: 10.1214/20-EJP470


Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius $R$, normalised by area, converges to a constant as $R\to \infty $. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals $c_{ES}(\ell )$ and $c_{LS}(\ell )$ that encode the density of excursion/level set components at the level $\ell $. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result, which derives differentiability of the functionals from the decay of the probability of ‘four-arm events’ for the field conditioned to have a saddle point at the origin. For some fields, including the important special cases of the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which allows us to deduce regions on which $c_{ES}(\ell )$ and $c_{LS}(\ell )$ are monotone.


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Dmitry Beliaev. Michael McAuley. Stephen Muirhead. "Smoothness and monotonicity of the excursion set density of planar Gaussian fields." Electron. J. Probab. 25 1 - 37, 2020.


Received: 15 July 2019; Accepted: 26 May 2020; Published: 2020
First available in Project Euclid: 11 August 2020

zbMATH: 07252725
MathSciNet: MR4136473
Digital Object Identifier: 10.1214/20-EJP470

Primary: 58K05, 60G15, 60G60


Vol.25 • 2020
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