Ecole de printemps du GDR MEGA 2024

9:30 - 10:45 (1h15)

Cours

Mini-cours de Laure Dumaz

10:45 - 11:00 (15min)

Pause café

11:00 - 12:15 (1h15)

Cours

Mini-cours de Thomas Leblé

12:15 - 14:00 (1h45)

Déjeuner

14:00 - 14:40 (40min)

Exposé Asymptotics in beta ensembles at high temperature

Exposé de Charlie Dworaczek Guera Abstract: In the celebrated beta-ensemble model, one can choose to pick the temperature proportional to the number of particles. In this regime, entropy plays a role in the determination of the equilibrium measure. We show CLT and asymptotics of the partition function.

14:40 - 15:00 (20min)

Exposé Random Matrix Approach to Tensor Approximation

Exposé de Hugo Lebeau Abstract: This work presents a comprehensive understanding of the estimation of a planted low-rank signal from a general spiked tensor model near the computational threshold. Relying on standard tools from the theory of large random matrices, we characterize the large-dimensional spectral behavior of the unfoldings of the data tensor and exhibit relevant signal-to-noise ratios governing the detectability of the principal directions of the signal. These results allow to accurately predict the reconstruction performance of truncated multilinear SVD (MLSVD) in the non-trivial regime. This is particularly important since it serves as an initialization of the higher-order orthogonal iteration (HOOI) scheme, whose convergence to the best low-multilinear-rank approximation depends entirely on its initialization. We give a sufficient condition for the convergence of HOOI and show that the number of iterations before convergence tends to $1$ in the large-dimensional limit.

15:00 - 15:15 (15min)

Pause

15:15 - 15:35 (20min)

Exposé De l’écologie aux matrices aléatoires

Exposé de Mohammed-Younes GUEDDARI Abstract: L’analyse du point d’équilibre dans un système Lotka-Volterra, qui régit les interactions complexes entre espèces, constitue un défi inhérent à la modélisation écologique. Notre recherche, se démarquant par son contexte de grande dimension, aborde cette problématique en utilisant un modèle de matrice d’interaction aléatoire de grande dimension [May72]. On établit également une connexion intrigante avec le Linear Complementarity Problem (LCP) [AHMN23] qui est un problème d’optimisation classique. Nous utilisons les algorithmes Approximate Message Passing (AMP) [FVRS21], reconnus comme des outils analytiques très puissants, pour étudier les propriétés statistiques du point d’équilibre. Notre travail se concentre sur l’extension de ces algorithmes, tra- ditionnellement appliqués à des matrices symétriques, afin de les adapter à des matrices non symétriques. Nous examinons plus particulièrement les matrices d’interaction ellip- tiques pour une modélisation plus réaliste des interactions complexes et asymétriques entre espèces au sein des écosystèmes. Si le temps le permet, je présenterai également des résultats de la propagation du chaos [Szn91]. En les appliquant à notre problème d’écologie théorique, ces résultats fourniront des insights plus fins sur les propriétés statistiques des sous-populations partageant des propriétés de croissance intrinsèques similaires.

15:35 - 15:55 (20min)

Exposé Chemical potential from correlation functions.

Exposé de Fabio Frommer Abstract: Interacting particle systems in a finite volume in equilibrium are often described by a grand canonical ensemble induced by the corresponding Hamiltonian, i.e. a finite volume Gibbs measure. However, in practice, directly measuring this Hamiltonian is not possible, as such, methods need to be developed to calculate the Hamiltonian potentials from measurable data. In this work we give an expansion of the chemical potential in terms of the correlation functions of such a system in the thermodynamic limit. This is a justification of a formal approach of Nettleton and Green from the 50's

15:55 - 16:10 (15min)

Pause

16:10 - 16:30 (20min)

Exposé Gaussian fluctuation for the Elliptic Ginibre Ensemble

Exposé Sylvain Chabredier Abstract. The elliptic Ginibre ensemble and the ground state of a free Fermi gas associated with a quadratic potential are two linked determinantal point processes for which the kernel is known exactly. In this talk, I will show recent results that we obtained jointly with Gaultier Lambert, describing the gaussian fluctuations of these point processes. For the Elliptic Ginibre Ensemble, I will present a uniform central limit theorem valid for linear statistics of the points (uniform for the correlation parameter). For the free Fermi gas, I will present the covariance structure associated with the gaussian fluctuations.

16:30 - 16:50 (20min)

Exposé correlation functions in Sinh-Gordon 1+1 QFT

Exposé de Alex Simon Résumé:The 1+1d Sinh-Gordon model is one of the simplest integrable models, but one has yet to construct it as a proper quantum field theory. One way to do it is to study its correlation functions and to make sure these functions are well-defined as series of N-uple integrals. They also have to satisfy several conditions, which are the axioms of Wightman. Here, we establish the well-definition of some truncated multi-point correlation functions and their properties.

Programme