Smoothness and monotonicity of the excursion set density of planar Gaussian fields

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http://hdl.handle.net/10138/319172

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Beliaev , D , McAuley , M & Muirhead , S 2020 , ' Smoothness and monotonicity of the excursion set density of planar Gaussian fields ' , Electronic Journal of Probability , vol. 25 , 93 . https://doi.org/10.1214/20-EJP470

Title: Smoothness and monotonicity of the excursion set density of planar Gaussian fields
Author: Beliaev, Dmitry; McAuley, Michael; Muirhead, Stephen
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2020
Language: eng
Number of pages: 37
Belongs to series: Electronic Journal of Probability
ISSN: 1083-6489
URI: http://hdl.handle.net/10138/319172
Abstract: 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 -> infinity. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals c(ES)(l) and c(LS)(l) that encode the density of excursion/level set components at the level l. 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)(l) and c(LS)(l) are monotone.
Subject: Gaussian fields
nodal set
level sets
critical points
PERCOLATION
NUMBER
111 Mathematics
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