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© 2018

Introduction to Random Matrices

Theory and Practice

Book

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 26)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 1-5
  3. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 7-13
  4. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 15-21
  5. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 23-31
  6. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 33-43
  7. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 45-51
  8. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 53-56
  9. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 57-63
  10. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 65-66
  11. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 67-74
  12. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 75-79
  13. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 81-87
  14. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 89-95
  15. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 97-103
  16. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 105-108
  17. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 109-117
  18. Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 119-124

About this book

Introduction

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 analytical results.

Keywords

Coulomb Gas Approach Diagrammatic Method Edwards-Jones Replica Theory Gaussian Matrices Random Matrices with Real Spectrum Resolvent Method Tracy-Widom Law Wigner's Semicircle Law Wishart Matrices Wigner surmise Joint probability density function of eigenvalues Jpdf of eigenvalues Vandermonde matrix Vandermonde determinant Andreief indentity Wishart-Laguerre ensemble Marcenko-Pastur distribution Gaussian Orthogonal Ensemble Edwards-Jones formalism Dyson Coulomb gas

Authors and affiliations

  1. 1.Department of Computer ScienceUniversity College LondonLondonUnited Kingdom
  2. 2.Instituto de FísicaUniversidade Federal de UberlândiaUberlândiaBrazil
  3. 3.Department of MathematicsKing’s College LondonLondonUnited Kingdom

Bibliographic information

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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)