One day workshop on Random Fields

Speakers

Program

Valentina Cammarota, Università di Roma La Sapienza

Alessia Caponera, LUISS Guido Carli

Guillaume Poly, Université de Rennes 1

Nicola Turchi, Università di Milano-Bicocca

Hugo Vanneuville, Institut Fourier (Université Grenoble Alpes)

9:45-10:00    Welcome

10:00-10:45   Poly

10:45-11:15    Coffee Break 

11:15-12:00    Caponera

12:00-12:45   Vanneuville

12:45-15:30    Free Time 

15:30-16:15    Cammarota

16:15-17:00    Turchi

19:30-               Social dinner 

Valentina Cammarota

Title: Correlation structure and resonant pairs for arithmetic random waves

Abstract: The geometry of Arithmetic Random Waves has been extensively investigated in the last fifteen years. In this talk we will discuss the correlation structure among different functionals such as nodal length, boundary length of excursion sets, and the number of intersection of nodal sets with deterministic curves in different classes; the amount of correlation depends in a subtle fashion from the values of the thresholds considered and the symmetry properties of the deterministic curves. In particular, we prove the existence of resonant pairs of threshold values where the asymptotic correlation is full, that is, at such values one functional can be perfectly predicted from the other in the high energy limit. We focus mainly on the 2-dimensional case but we discuss some specific extensions to dimension 3. Joint work with R. Maffucci, D. Marinucci and M. Rossi. 

Alessia Caponera

Title: Multiscale inference for time-dependent spherical random fields

Abstract: The analysis of time-dependent spherical random fields is the natural setting for a number of different areas of applications, such as Cosmology, Astrophysics, Geophysics, Climate and Atmospheric Science. In these areas, it is often a valid question to probe whether structural breaks have occurred over time; the most immediate example of such changes is obviously represented by shifts in the global mean, which would correspond to Global Warming when studying temperature data. We then present some inferential tools to study the behaviour of structural breaks in the harmonic domain (allowing modifications which may go beyond a simple global mean shift). Our approach, which intrinsically integrates the spatial and temporal dimensions, could give multiscale insights into both the global and local behaviour of changes. This will be motivated by a real dataset of global surface temperature anomalies.

Guillaume Poly

Title: About the CLT for linear statistics of beta ensembles and the phenomenon of «superconvergence»

Abstract: We will introduce the classical model of Beta ensembles as well as its connection with random matrices. We will be more particularly interested in the CLT for the so-called linear statistics. We will adopt the point of view of «Stein’s method» and «Gamma calculus», two techniques that I will introduce. Our goal will be to provide a simple and quantitative proof which gives better estimates (both in term of speed of convergence and required regularity) than recent advances on this subject by (Beckerman-Leblé-Serfaty) and (Lambert-Ledoux-Webb). We shall also discuss the question of the metric of convergence and will exhibit a regularization phenomenon enabling to upgrade the classical mode of convergence (Wasserstein) to the uniform convergence of the density and all its derivatives on the real line. If time permits, we will review some recent results related to this mode of convergence for Gaussian polynomials in relation with problems of anti-concentration and random graph theory. Joint work with Jurgen Angst and Ronan Herry. 

Nicola Turchi

Title: Beta random polytopes in high dimension: phase transition of the expected volume

Abstract: Beta polytopes are a class of random polytopes, which arise as convex hulls of independent random points distributed according to a certain radially-symmetric probability distribution supported on the Euclidean ball, called the beta distribution. Examples are the uniform distribution in the ball and the uniform distribution on the sphere. As the space dimension grows, the expected fraction of the volume that these polytopes fill within their ambient balls can be asymptotically negligible or not, depending on the number of points which are chosen in each dimension. In this talk we give an overview on how to quantify this statement, first showing a rough threshold for the aforementioned growth and secondly a more precise one, namely how many points are needed to get any fraction of the volume in average. Lastly, we show how we can handle more precise asymptotics in the lower regimes of points. Other intrinsic volumes are also discussed. Based on works with G. Bonnet and Z. Kabluchko.

Hugo Vanneuville

Title: Are there unbounded nodal lines?

Abstract: In this talk, we will study several examples of smooth Gaussian fields f from R^d to R (with d at least 2) and we are going to ask if there are unbounded component in the nodal set {f=0}. Several questions will arise: Can we say things when the covariance function is not always positive? Can discrete percolation theory help? On the contrary, can the continuous structure of the model provide new tools/new ideas? Joint works with Stephen Muirhead and Hugo Duminil-Copin, Alejandro Rivera and Pierre-François Rodriguez.