Random Matrix Theory with an External Source

  • Edouard Brézin
  • Shinobu Hikami

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 19)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Edouard Brézin, Shinobu Hikami
    Pages 1-2
  3. Edouard Brézin, Shinobu Hikami
    Pages 3-15
  4. Edouard Brézin, Shinobu Hikami
    Pages 17-23
  5. Edouard Brézin, Shinobu Hikami
    Pages 25-35
  6. Edouard Brézin, Shinobu Hikami
    Pages 37-60
  7. Edouard Brézin, Shinobu Hikami
    Pages 61-64
  8. Edouard Brézin, Shinobu Hikami
    Pages 65-98
  9. Edouard Brézin, Shinobu Hikami
    Pages 99-111
  10. Edouard Brézin, Shinobu Hikami
    Pages 113-121
  11. Edouard Brézin, Shinobu Hikami
    Pages 123-129
  12. Back Matter
    Pages 131-138

About this book


This is a first book to show that the theory of the Gaussian random matrix is essential to understand the universal correlations with random fluctuations and to demonstrate that it is useful to evaluate topological universal quantities. We consider Gaussian random matrix models in the presence of a deterministic matrix source. In such models the correlation functions are known exactly for an arbitrary source and for any size of the matrices. The freedom given by the external source allows for various tunings to different classes of universality. The main interest is to use this freedom to compute various topological invariants for surfaces such as the intersection numbers for curves drawn on a surface of given genus with marked points, Euler characteristics, and the Gromov–Witten invariants. A remarkable duality for the average of characteristic polynomials is essential for obtaining such topological invariants. The analysis is extended to nonorientable surfaces and to surfaces with boundaries.


Random matrix theory Gaussian random matrix models 2D quantum gravity Kontsevich Airy matrix model Gromov-Witten invariants

Authors and affiliations

  • Edouard Brézin
    • 1
  • Shinobu Hikami
    • 2
  1. 1.Laboratoire de Physique ThéoriqueÉcole Normale SupérieureParisFrance
  2. 2.Mathematical and Theoretical Physics UnitOkinawa Institute of Science and Technology Graduate UniversityKunigami-gunJapan

Bibliographic information

  • DOI
  • Copyright Information The Author(s) 2016
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-981-10-3315-5
  • Online ISBN 978-981-10-3316-2
  • Series Print ISSN 2197-1757
  • Series Online ISSN 2197-1765
  • Buy this book on publisher's site
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