Random matrix theory has found applications in many fields of physics (disordered systems, stability of dynamical systems, interface models, electronic transport,...) and mathematics (operator algebra, enumerative combinatorics, number theory,...). A recurrent problem in many domains is to understand how the spectra of two random matrices recombine...
Large random matrices have been proved to be of fundamental importance in mathematics (high dimensional probability, operator algebras, combinatorics, number theory,...) and in physics (nuclear physics, quantum fields theory, quantum chaos,..) for a long time. The use of large random matrices is more recent in statistical signal processing and time...
This statistical physics thesis focuses on the study of three kinds of systems which display repulsive interactions: eigenvalues of random matrices, non-crossing random walks and trapped fermions. These systems share many links, which can be exhibited not only at the level of their static version, but also at the level of their dynamical version. W...
This thesis deals with the optimal transport problem, in particular with regularity properties shared by optimal transport maps. The first part of this manuscript provides a new proof of the Caffarelli contraction theorem, stating that the optimal transport map from the gaussian measure to a measure with a uniformly log-concave density with respect...
This PhD lies at the intersection of Random Matrix Theory and Free Probability Theory. The connection between those two fields dates back to the early nineties with the work of Voiculescu who created the Theory of free probability. A probability theory for noncommutative variables where the notion of freeness replaces the one of independence in cla...
This thesis is motivated by the study of covariance matrices, and is naturally structured in three parts. In the first part, we study dynamic models related to covariance matrices. More precisely, we study the systems of stochastic differential equations inherited from the dynamics of the eigenvalues of matrix valued processes named the Wishart pro...
Cette thèse est motivée par l'étude des matrices de covariance, et s'articule naturellement en trois parties. Dans la première partie, nous étudions des modèles dynamiques liés aux matrices de covariance. Nous étudions plus précisément les systèmes d'équations différentielles stochastiques hérités de la dynamique des valeurs propres de processus ma...
This thesis tackles portfolio allocation problems in studying robust covariance matrix estimators and the dynamic dependence between financial assets to improve the overall performance of risk-based allocation strategies. In the first part of this thesis, we focus on the covariance matrix estimation and we develop for it a robust and de-noised esti...
A number of recent works proposed to use large random matrix theory in the context of high-dimensional statistical signal processing, traditionally modeled by a double asymptotic regime in which the dimension of the time series and the sample size both grow towards infinity. These contributions essentially addressed detection or estimation schemes ...
The topic of this thesis is the development of large-scale optimization algorithms, popularized by machine learning, for various applications around light propagation through complex media. We study these questions through the prism of optical scattering, a physical phenomenon describing the propagation of light in complex materials. This unconvent...
