Les modèles classiques de matrices aléatoires sont apparus en physique il y a quelques années; les probabilistes en ont par la suite étudié différents aspects (lois des valeurs propres, processus associés aux valeurs propres). La recherche actuelle comporte de nombreuses facettes, allant de la géométrie à l’analyse fonctionnelle. On trouve par ailleurs des modèles de matrices aléatoires en biologie et en statistique. Le cours va consister à donner les bases mathématiques de la théorie des matrices aléatoires, et de présenter de nouvelles questions qui se posent actuellement dans diverses applications (réseaux de communication sans fil ou statistique).
Charles Bordenave (University of Toulouse): Spectrum of sparse random graphs
Unimodular graphs are the natural limits of finite graph sequences with uniformly bounded degrees. For these graphs, we will define a proper notion of spectrum and study its properties. We will put the emphasis on the study of the regularity of the spectrum and the study of its atoms. Our main examples will come from percolation graphs and classical random graphs. We will also discuss some connections with random matrices.
Alice Guionnet (MIT and ENS Lyon): Basics on Random Matrices
In this course we will discuss the properties of Wigner matrices, that is Herimtian matrices with independent entries modulo the symmetry constraint. We will first investigate the macroscopic properties such as the mean asymptotic behaviour of the eigenvalues, and then the local properties such as the fluctuations of the spacing between two eigenvalues. We will finally review some open problems.
Jamal Najim (University of Marne La Vallée): Large random matrices and applications to statistics and electrical engineering
In this course, we shall present random matrix models of interest in electrical engineering and statistics and study their properties in the large dimensional regime. After the mathematical framework is set up, we will describe applications of large random matrices to wireless communication, statistical signal processing and statistics. We will also present areas of current interest where the presented techniques are relevant. |