Introduction to Random Matrices - Theory and Practice

TL;DR

This book provides an accessible, beginner-friendly introduction to random matrix theory, covering classical and modern topics with detailed explanations and numerical verifications for physicists and mathematicians starting their research.

Contribution

It offers a comprehensive, easy-to-understand introduction to RMT, including advanced topics like replica approach and free probability, tailored for newcomers.

Findings

Clear explanations of classical ensembles and spectral properties

Inclusion of modern topics like replica approach and free probability

Emphasis on numerical verification of analytical results

Abstract

This is a book for absolute beginners. If you have heard about random matrix theory, commonly denoted RMT, but you do not know what that is, then welcome!, this is the place for you. Our aim is to provide a truly accessible introductory account of RMT for physicists and mathematicians at the beginning of their research career. We tried to write the sort of text we would have loved to read when we were beginning Ph.D. students ourselves. Our book is structured with light and short chapters, and the style is informal. The calculations we found most instructive are spelt out in full. Particular attention is paid to the numerical verification of most analytical results. Our book covers standard material - classical ensembles, orthogonal polynomial techniques, spectral densities and spacings - but also more advanced and modern topics - replica approach and free probability - that are not normally included in elementary accounts on RMT. This book is dedicated to the fond memory of Oriol Bohigas.