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

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 $R\to \infty$. 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.

arXiv comment39 pages, 8 figures. Updated to match published version

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Cited by in corpus (4)

• Fluctuations of the number of excursion sets of planar Gaussian fields
• A central limit theorem for the number of excursion set components of Gaussian fields
• A covariance formula for the number of excursion set components of Gaussian fields and applications
• Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension $d\ge 3$