Introduction to Random Matrices: Theory and Practice

by Giacomo Livan, Marcel Novaes, Pierpaolo Vivo

Overview

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.

Product Details

ISBN-13:	9783319708836
Publisher:	Springer International Publishing
Publication date:	01/17/2018
Series:	SpringerBriefs in Mathematical Physics , #26
Edition description:	1st ed. 2018
Pages:	124
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

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 Marčenko and Pastur.- Replicas.- Replicas for GOE.- Born to be free.

Editorial Reviews

The text is well written, and the authors’ informal conversational style sets the book up nicely for someone who is using it as a self-study guide in the area. The chapters are short, but there are numerous examples, and the most instructive calculations are represented in full. Particular attention is paid to the numerical verification of the most analytical results.” (Susanna Spektor, Mathematical Reviews, September, 2018)

“Giacomo Livan, Marcel Novaes, and Pierpaolo Vivo have written a small but lively book on random matrices. … The book feels conversational. It has an informal tone, a sense of humor, and a pleasant mixture of exposition and detail. The pace is brisk, appropriate for a book aiming to be an introductory survey rather than a formal textbook.” (John D. Cook, MAA Reviews, July, 2018)

“The emphasis is on concepts, computations and tricks of the trade, and the style is modern and informal, which the reader will no doubt enjoy. … This monograph should prove to become a very welcome companion to the serious and enthusiastic reader setting out to get to know this vast and very useful subject.” (Rabe von Randow, zbMATH 1386.15003, 2018)

From the Publisher