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Random Matrix Theory with an External Source

  • Book
  • © 2016

Overview

Authors:
  1. Edouard Brézin

    1. Laboratoire de Physique Théorique, École Normale Supérieure, Paris, France

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  2. Shinobu Hikami

    1. Mathematical and Theoretical Physics Unit, Okinawa Institute of Science and Technology Graduate University, Kunigami-gun, Japan

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  • Expresses the correlation function of the Gaussian random matrix model with an external source in the integral formula
  • Examines universal behaviors of level spacing distributions for an arbitrary external source
  • Obtains the topological invariants such as the intersection numbers of the moduli space of spin curves by the duality and replica method in the scaling limit of the external source
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Edouard Brézin, Shinobu Hikami
    Pages 1-2

  3. Gaussian Means

    • Edouard Brézin, Shinobu Hikami
    Pages 3-15

  4. External Source

    • Edouard Brézin, Shinobu Hikami
    Pages 17-23

  5. Characteristic Polynomials and Duality

    • Edouard Brézin, Shinobu Hikami
    Pages 25-35

  6. Universality

    • Edouard Brézin, Shinobu Hikami
    Pages 37-60

  7. Intersection Numbers of Curves

    • Edouard Brézin, Shinobu Hikami
    Pages 61-64

  8. Intersection Numbers of p-Spin Curves

    • Edouard Brézin, Shinobu Hikami
    Pages 65-98

  9. Open Intersection Numbers

    • Edouard Brézin, Shinobu Hikami
    Pages 99-111

  10. Non-orientable Surfaces from Lie Algebras

    • Edouard Brézin, Shinobu Hikami
    Pages 113-121

  11. Gromov–Witten Invariants, P\(^{1}\) Model

    • Edouard Brézin, Shinobu Hikami
    Pages 123-129

  12. Back Matter

    Pages 131-138
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Keywords

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.

Authors and Affiliations

  • Laboratoire de Physique Théorique, École Normale Supérieure, Paris, France

    Edouard Brézin

  • Mathematical and Theoretical Physics Unit, Okinawa Institute of Science and Technology Graduate University, Kunigami-gun, Japan

    Shinobu Hikami

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