Excursion sets of planar Gaussian fields
Keywords: Excursion sets
Critical points
Gaussian fields
Random fields
Level sets

Gaussian fields are widely used for modelling spatial phenomena in disciplines such as cosmology, medical imaging and quantum mechanics. Geometric properties of the excursion sets of such fields are well studied, as they provide important test statistics in these areas. There is also theoretical interest in the large scale topological behaviour of excursion sets of Gaussian fields, due to recent results suggesting a connection with discrete percolation models. In this thesis we study the geometry of planar Gaussian fields with a focus on one particular quantity; the number of excursion/level set components in a large domain, which is of interest both for statistical testing and in relation to percolation.
Chapter 1 introduces the topic of excursion/level sets of Gaussian fields. This includes motivation for their study, an outline of some mathematical techniques for their analysis and a summary of the relevant literature.
In Chapter 2 we consider the mean number of connected components of a Gaussian excursion set or level set in a large domain. Nazarov and Sodin have shown that this quantity satisfies a law of large numbers as the size of the domain increases. The value of the corresponding limit, at different levels, can be viewed as a functional which describes the asymptotic density of excursion set components. We derive a new integral representation of this functional, which provides an alternative way of analysing it. The representation is based on a deterministic decomposition of excursion set components in terms of critical points, which is analogous to the Poincaré-Hopf theorem for the Euler characteristic. We also derive some basic properties of the excursion set functional, including symmetries, upper/lower bounds and continuity properties. Finally, we explicitly compute this functional for a simple class of fields, which poses some interesting questions about the general properties of the functional.
In Chapter 3, we extend the analysis of our integral representations to show that the mean functionals are continuously differentiable for a wide class of fields. This follows from a more general result, in which we derive 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. We discuss how this compares to related results for percolation models. For some fields, including two important examples known as the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which proves that the mean functionals are monotone, with respect to the level, on certain regions.
In Chapter 4 we give the most significant result of this thesis: a lower bound on the variance of the number of Gaussian excursion/level set components in a large domain. More precisely, we prove a lower bound on the order of the fluctuations of these variables, for certain classes of field and certain levels, which implies the corresponding variance bound. These bounds are conjectured to be optimal for generic fields, based on the behaviour of other geometric quantities and also on the corresponding properties of discrete percolation models. The bounds are intuitively driven by `level shifts' of the field and are proven by comparing the number of components at different levels, which relies heavily on the results from Chapters 2 and 3.