Aspects of random matrix theory: Concentration and subsequence problems.

摘要

The present work studies some aspects of random matrix theory. Its first part is devoted to the asymptotics of random matrices with infinitely divisible, in particular heavy-tailed, entries. Its second part focuses on relations between limiting law in subsequence problems and spectra of random matrices. In Chapter II, we give concentration inequalities for the spectral measure, with respect to the Wasserstein distance, or for the maximal eigenvalue of random Hermitian matrices with infinitely divisible (not necessarily independent) entries. For such matrices, the classical techniques, which rely on the independence and/or the finite moments properties of the entries, no longer apply. Results for the spectral measure of matrices with stable entries are also obtained leading to different rates of decay for different ranges of deviation. Finally, concentration results for various functions of Wishart matrices are also derived. In Chapter III and Chapter IV, interactions between spectra of classical Gaussian ensembles and subsequence problems are studied with the help of the powerful machinery of Young tableaux. For the random word problem, from an ordered finite alphabet, the shape of the associated Young tableaux is shown to converge to the spectrum of the (generalized) traceless GUE. Various properties of the (generalized) traceless GUE are established, such as a law of large numbers for the extreme eigenvalues and the convergence of the spectral measure towards the semicircle law. The limiting shape of the whole tableau is also obtained as a Brownian functional. The Poissonized word problem is finally discussed, and, using Poissonization, the convergence of the whole Poissonized tableaux is derived.