Abstract
Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic compact manifolds. Our work is motivated by questions about the geometry of such random functions, in particular relating to the structure of their nodal and level sets. We study four discretisation schemes that extract information about level sets of planar Gaussian fields. Each scheme recovers information up to a different level of precision, and each requires a maximum mesh-size in order to be valid with high probability. The first two schemes are generalisations and enhancements of similar schemes that have appeared in the literature (Beffara and Gayet in Publ Math IHES, 2017. https://doi.org/10.1007/s10240-017-0093-0; Mischaikow and Wanner in Ann Appl Probab 17:980–1018, 2007); these give complete topological information about the level sets on either a local or global scale. As an application, we improve the results in Beffara and Gayet (2017) on Russo–Seymour–Welsh estimates for the nodal set of positively-correlated planar Gaussian fields. The third and fourth schemes are, to the best of our knowledge, completely new. The third scheme is specific to the nodal set of the random plane wave, and provides global topological information about the nodal set up to ‘visible ambiguities’. The fourth scheme gives a way to approximate the mean number of excursion domains of planar Gaussian fields.
Highlights
Let Ψ : R2 → R be a planar Gaussian field, that is, a random function whose finitedimensional distributions are Gaussian random variables
We shall throughout this paper assume that Ψ is stationary and normalised to have zero mean and unit variance at each point. This implies that Ψ may be defined through its positive-definite correlation kernel κ : R2 → [−1, 1], satisfying κ(0) = 1 and, for each s, t ∈ R2, κ(s − t) := E[Ψ (s)Ψ (t)]
We refer to the components of N as the level lines and the components of R2 \ N as the excursion domains
Summary
Let Ψ : R2 → R be a planar Gaussian field, that is, a random function whose finitedimensional distributions are Gaussian random variables. An early work was [11], which developed such a scheme for general planar random fields This is very similar to our first scheme—assessing the validity of the discretisation on the local scale, see Theorem 2—and is based around controlling the event that the level set N crosses an edge twice For each lattice L there exists a visibility parameter μ > 0 such that we may define a certain set of Type 2 error patterns on collections of faces that lie inside balls of radius μ, for which the discretisation up to ambiguities holds by analogy to Theorem 5.
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