On the number of excursion sets of planar Gaussian fields

The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

On the excursion area of perturbed Gaussian fields

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ2 under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.

Searching cosmic strings network in the CMB

The CMB stochastic field provides us with a unique opportunity to search for the predicted imprints of various cosmological theories. In this work, we explore the detectability of the cosmic string (CS) network traces in the pixel-space through its gravitational impact on the CMB anisotropies, i.e. the Gott–Kaiser–Stebbins effect. First, in a classical approach, we use a series of multi-scale edge-detection algorithms followed by certain critical and excursion sets measures on the CMB simulated data (with different levels of contributions from the CSs). With this pipeline the minimum value of detectable CS’s tension for noiseless maps with a resolution of [Formula: see text] is found to be [Formula: see text]. Calling for the help the power of the machine learning algorithms with our proposed feature vectors, decreases the above lower bound to [Formula: see text]. The methods developed and used in this work are quite general and applicable to a wide variety of pattern search problems in the various stochastic fields ranging from cosmological random fields to other natural and artificial complex systems.