École Nationale Supérieure de
Techniques Avancées

Abstract

The study of eigenvalues is fundamental in random matrix theory. This interest finds its roots in problems of quantum physics in which the eigenvalues of the Hamiltonian operator correspond to the set of possible outcomes of the energy of a system. It is known that if a projection of rank n-1 is applied to a full rank hermitian matrix of size n, the eigenvalues of the projected matrix and the eigenvalues of the full ranked matrix are interlaced (Cauchy Interlacing theorem \cite{cauchy}). By iterating on the rank of successive projections, a sequence of projections called a flag is obtained. Under certain conditions on the starting random hermitian matrix, the set of successive eigenvalues obtained is determinantal. The interest of a determinantal structure is that to calculate joint expectations, instead of integrating correlation functions, we can calculate the determinant of a certain operator called the kernel of the process. Therefore having a determinantal process enables to deal with probabilistic calculus more easily. The purpose of the internship is to study the following question : "If two flags are considered simultaneously, are the successive eigenvalues still determinantal ?".