Introduction to Random Matrices

Modern developments of Random Matrix Theory as well as pedagogical approaches to the standard core of the discipline are surprisingly hard to find in a well-organized, readable and user-friendly fashion. This slim and agile book, written in a pedagogical and hands-on style, without sacrificing formal rigor fills this gap. It brings Ph.D. students in Physics, as well as more senior practitioners, through the standard tools and results on random matrices, with an eye on most recent developments that are not usually covered in introductory texts. The focus is mainly on random matrices with real spectrum.The main guiding threads throughout the book are the Gaussian Ensembles. In particular, Wigner's semicircle law is derived multiple times to illustrate several techniques (e.g., Coulomb gas approach, replica theory).Most chapters are accompanied by Matlab codes (stored in an online repository) to guide readers through the numerical check of most analyticalresults.

Includes supplementary material: sn.pub/extras

Inhalt
Preface.- Getting Started.- Value the eigenvalue.- Classified Material.- The fluid semicircle.- Saddle-point-of-view.- Time for a change.- Meet Vandermonde.- Resolve(nt) the semicircle.- One pager on eigenvectors.- Finite N.- Meet Andréief.- Finite N is not finished.- Classical Ensembles:Wishart-Laguerre.- Meet Marenko and Pastur.- Replicas.- Replicas for GOE.- Born to be free.