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.
Highlights
Let f : R2 → R be a continuous stationary Gaussian field with zero mean and covariance function K : R2 → R defined by K(x) = E(f (x)f (0))
We are interested in the geometric properties of theexcursion sets and level sets of this field, defined respectively as x ∈ R2 : f (x) ≥ and x ∈ R2 : f (x) =
As a particular example; cosmological theories predict that the Cosmic Microwave Background Radiation observed on Earth can be well modelled as a realisation of a stationary Gaussian field on the two-dimensional sphere
Summary
This can be expressed as a power series in p, and the smoothness of κ can be deduced from bounds on the coefficients in terms of connection probabilities for the cluster at the origin This approach does not readily generalise to the setting of Gaussian fields: whilst it can be shown that cES ( ) = E Vol(C)−11f(0)> , where Vol(C) is the volume of the component of x ∈ R2 : f (x) ≥ containing the origin, it is not known whether the density of (Vol(C), f (0)) is jointly continuous ([8] studies a kind of ‘ergodic’ density for Vol(C) at the zero level). Our study of the integral representation for cES and cLS allows us to derive certain montonicity properties of these functionals (see Propositions 2.20–2.22); these results are of independent interest, and are a key input to proving lower bounds on the variance of the number of excursion/level sets of Gaussian fields (see Remark 2.24)
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