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  • Book
  • © 2018

Introduction to Random Matrices

Theory and Practice

Authors:

  1. Giacomo Livan

    1. Department of Computer Science, University College London, London, United Kingdom

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    You can also search for this author in PubMed   Google Scholar

  2. Marcel Novaes

    1. Instituto de Física, Universidade Federal de Uberlândia, Uberlândia, Brazil

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    You can also search for this author in PubMed   Google Scholar

  3. Pierpaolo Vivo

    1. Department of Mathematics, King’s College London, London, United Kingdom

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    You can also search for this author in PubMed   Google Scholar

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

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  • ISBN: 978-3-319-70885-0
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Table of contents (17 chapters)

  1. Front Matter

    Pages i-ix
    PDF

  2. Getting Started

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 1-5

  3. Value the Eigenvalue

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 7-13

  4. Classified Material

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 15-21

  5. The Fluid Semicircle

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 23-31

  6. Saddle-Point-of-View

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 33-43

  7. Time for a Change

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 45-51

  8. Meet Vandermonde

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 53-56

  9. Resolve(nt) the Semicircle

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 57-63

  10. One Pager on Eigenvectors

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 65-66

  11. Finite N

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 67-74

  12. Meet Andréief

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 75-79

  13. Finite N Is Not Finished

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 81-87

  14. Classical Ensembles: Wishart-Laguerre

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 89-95

  15. Meet Marčenko and Pastur

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 97-103

  16. Replicas...

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 105-108

  17. Replicas for GOE

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 109-117

  18. Born to Be Free

    • Giacomo Livan, Marcel Novaes, Pierpaolo Vivo
    Pages 119-124
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About this book

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.
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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
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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)
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Authors and Affiliations

  • Department of Computer Science, University College London, London, United Kingdom

    Giacomo Livan

  • Instituto de Física, Universidade Federal de Uberlândia, Uberlândia, Brazil

    Marcel Novaes

  • Department of Mathematics, King’s College London, London, United Kingdom

    Pierpaolo Vivo

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Bibliographic Information

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Access via your institution
eBook USD 54.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-70885-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 69.99
Price excludes VAT (USA)
Learn about institutional subscriptions