Speakers

List of speakers (in alphabetic order) and abstracts

Cammarota, Valentina (La Sapienza, Rome)

Title: On critical points and excursion sets of random Laplace eigenfunctions on compact manifolds

Abstract: We study the critical points and the excursion sets of smooth Gaussian random fields on compact real-analytic manifolds (e.g. sphere and flat torus). We discuss the limiting distribution and asymptotic fluctuations of the number of critical points of random Laplace eigenfunctions in the high energy limit; this requires a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances. Building upon such results and Wiener-Itˆo chaos decomposition, we derive a Central Limit Theorem for the Euler characteristic of the excursion sets. Applications to data analysis in Cosmology are also discussed. Based on joint works with Domenico Marinucci (Universita` degli Studi di Roma Tor Vergata)and Igor Wigman (King’s College London).

Döbler, Christian (Université de Luxembourg)

Title: Universal Gaussian fluctuations and fourth moment theorems of homogeneous sums

Abstract: The fourth moment phenomenon for chaotic random variables on various spaces has been intensively studied since it was first discovered by Nualart and Peccati (2005) in a Gaussian framework. In particular, it has been established for eigenfunctions of diffusive Markov generators and, more recently, also in the non-diffusive framework of Poisson random measures. On the other hand, Nourdin, Peccati and Reinert (2019) found that Gaussian fluctuations of homogeneous multilinear forms in i.i.d. standard normal random variables are universal, in the sense that asymptotic normality still holds (up to some technicality) if one replaces the Gaussian variables with an arbitrary i.i.d. sequence which has the same first and second moments. Such a universality result has also been established for Poisson homogeneous sums as well as for homogeneous sums in i.i.d. sequences which satisfy certain moment conditions. Moreover, it is known that for Rademacher homogeneous sums, neither the fourth moment theorem nor Gaussian universality holds. In this talk we review known results about universality and fourth moment theorems for homogeneous sums and also present some open problems and conjectures about these topics.

Krokowski, Kai (Ruhr-Universität Bochum)

Title: On the fourth moment condition for Rademacher chaos

Abstract: Adapting the spectral viewpoint suggested by Ledoux in the context of symmetric Markov diffusion generators and recently exploited in the non-diffusive setup of a Poisson random measure by Döbler and Peccati, we investigate the fourth moment condition for discrete multiple integrals with respect to general, i.e., non-symmetric and non-homogeneous, Rademacher sequences and show that, in this situation, the fourth moment alone does not govern the asymptotic normality. Indeed, here one also has to take into consideration the maximal influence of the corresponding kernel functions. In particular, we show that there is no exact fourth moment theorem for discrete multiple integrals of order m ≤ 2 with respect to a symmetric Rademacher sequence.

Lachièze-Rey, Raphaël (Université Paris Descartes)

Title: Excursions of shot noise fields, and non-stabilizing functionals

Abstract: Shot noise fields are random fields obtained by summing the independent contributions of random kernels centred on an underlying set of random points. An excursion is a random set obtained by thresholding the field above some prescribed value. We present asymptotic results on the variance and asymptotic normality for the volume and perimeter of such excursions sets, under homogeneous Poisson input. Since these functionals do not satisfy the classical hypothesis of "geometric stabilisation", we also present results that are supposed to deal with a general class of non-stabilising geometric functionals.

Rossi, Maurizia (Université Paris Descartes)

Title: Random Nodal Length and Wiener Chaos

Abstract: In this talk we present some of the recent results on the « nodal geometry » of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace eigenfunctions on the torus (arithmetic random waves) and on the sphere (random spherical harmonics). We give some insight on both Berry's cancellation phenomenon and the nature of nodal length second order fluctuations (non-Gaussian on the torus and Gaussian on the sphere) in terms of chaotic components. Finally we consider the general case of monochromatic random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian surface with frequencies from a short interval, whose scaling limit is Berry's Random Wave Model. For the latter we present some recent results on the asymptotic distribution of its nodal length in the high energy limit (equivalently, for growing domains).

Todino, Anna Paola (Gran Sasso Science Institute)

Title: Some results about the asymptotic distribution of the area of the excursion regions of random eigenfunctions on a spherical cap

Abstract: In this talk, we analyse the asymptotic behaviour of the excursion area of random Gaussian spherical harmonics in spherical caps. It is found that the second chaos component is the dominant term of the expansion of the variance. So, it is established a Quantitative Central Limit Theorem for the area, in the high energy limit. Moreover, we introduce a new infinitely differentiable function converging to the characteristic function in the space of integrable functions on the sphere.

Vidotto, Anna (Université de Luxembourg)

Title: Fourth Moment Theorems on the Poisson space in any dimension

Abstract: In a recent paper by C. D ̈obler and G. Peccati (2017), an exact quantitative fourth moment theorem (FMT) for multiple Wiener-Itˆo integrals on the Poisson space is established. In particular, the authors extended the spectral framework initiated by Ledoux (2012) from the situation of a diffusive Markov generator to the non-diffusive O-U generator on the Poisson space. On the other hand, very recently, I. Nourdin and G. Zheng (2017) reproved FMTs in the Gaussian setting via an innovative exchangeable pairs couplings construction. In this talk, we will show how we obtained, by adapting the exchangeable pairs couplings construction to the Poisson framework, an optimal improvement of the quantitative FMT in the univariate case, proving it under the weakest possible assumptions of finite fourth moments, as well as an extension of it to any dimension, namely a Peccati-Tudor type theorem.

Zheng, Guangqu (Université de Luxembourg)

Title: Fourth moment phenomena via exchangeable pairs

Abstract: In this talk, I will present recent application of elementary exchangeable pair couplings for obtaining the fourth moment theorems on chaotic structures. Some application towards Wishart matrices will be discussed if time permits.