It is well-known that a large class of determinantal processes including the sine-process satisfies the Central Limit Theorem. For many dynamical systems satisfying the CLT the Donsker Invariance Principle also takes place. The latter states that, in some appropriate sense, trajectories of the system can be approximated by trajectories of the Brownian motion. I will present results of my joint work with A. Bufetov, where we prove a functional limit theorem for the sine-process, which turns out to be very different from the Donsker Invariance Principle. We show that the anti-derivative of our process can be approximated by the sum of a linear Gaussian process and small independent Gaussian fluctuations whose covariance matrix we compute explicitly.[-]

Planar maps are planar graphs embedded in the sphere viewed modulo continuous deformations. There are two families of bijections between planar maps and lattice paths that are applied to prove scaling limit results of planar maps to so-called Liouville quantum gravity surfaces: metric bijections and mating-of-trees bijections. We will present scaling limit results obtained in this way, including works with Bernardi and Sun and with Albenque and Sun.[-]

I will discuss second order results for the length of nodal sets and the number of phase singularities associated with Gaussian random Laplace eigenfunctions, both on compact manifolds (the flat torus) and on subset of the plane. I will mainly focus on 'cancellation phenomena' for nodal variances in the high-frequency limit, with specific emphasis on central and non-central second order results.

Based on joint works with F. Dalmao, D. Marinucci, I. Nourdin, M. Rossi and I. Wigman.[-]

Anomaly detection in random fields is an important problem in many applications including the detection of cancerous cells in medicine, obstacles in autonomous driving and cracks in the construction material of buildings. Scan statistics have the potential to detect local structure in such data sets by enhancing relevant features. Frequently, such anomalies are visible as areas with different expected values compared to the background noise where the geometric properties of these areas may depend on the type of anomaly. Such geometric properties can be taken into account by combinations and contrasts of sample means over differently-shaped local windows. For example, in 2D image data of concrete both cracks, which we aim to detect, as well as integral parts of the material (such as air bubbles or gravel) constitute areas with different expected values in the image. Nevertheless, due to their different geometric properties we can define scan statistics that enhance cracks and at the same time discard the integral parts of the given concrete. Cracks can then be detected using asuitable threshold for appropriate scan statistics. 9 In order to derive such thresholds, we prove weak convergence of the scan statistics towards a functional of a Gaussian process under the null hypothesis of no anomalies. The result allows for arbitrary (but fixed) dimension, makes relatively weak assumptions on the underlying noise, the shape of the local windows and the combination of finitely-many of such windows. These theoretical findings are accompanied by some simulations as well as applications to semi-artifical 2D-images of concrete.[-]

We consider the monogenic representation for self-similar random fields. This approach is based on the monogenic representation of a greyscale image, using Riesz transform, and is particularly well-adapted to detect directionality of self-similar Gaussian fields. In particular, we focus on distributions of monogenic parameters defined as amplitude, orientation and phase of the spherical coordinates of the wavelet monogenic representation. This allows us to define estimators for some anisotropic fractional fields. We then consider the elliptical monogenic model to define vector-valued random fields according to natural colors, using the RGB color model. Joint work with Philippe Carre (XLIM, Poitiers), Céline Lacaux (LMA, Avignon) and Claire Launay (IDP, Tours).[-]