Random Matrices

Revised and Enlarged Second Edition

Madan Lal Mehta

• Publisher: Academic Press
• Year: 2014
• Print ISBN: 9780124880511
• eBook ISBN: 9781483295954
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Linear Algebra

Since the publication of Random Matrices (Academic Press, 1967) so many new results have emerged both in theory and in applications, that this edition is almost completely revised to reflect the developments. For example, the theory of matrices with quaternion elements was developed to compute certain multiple integrals, and the inverse scattering theory was used to derive asymptotic results. The discovery of Selberg's 1944 paper on a multiple integral also gave rise to hundreds of recent publications. This book presents a coherent and detailed analytical treatment of random matrices, leading in particular to the calculation of n-point correlations, of spacing probabilities, and of a number of statistical quantities. The results are used in describing the statistical properties of nuclear excitations, the energies of chaotic systems, the ultrasonic frequencies of structural materials, the zeros of the Riemann zeta function, and in general the characteristic energies of any sufficiently complicated system. Of special interest to physicists and mathematicians, the book is self-contained and the reader need know mathematics only at the undergraduate level.Key Features* The three Gaussian ensembles, unitary, orthogonal, and symplectic; their n-point correlations and spacing probabilities* The three circular ensembles: unitary, orthogonal, and symplectic; their equivalence to the Gaussian* Matrices with quaternion elements* Integration over alternate and mixed variables* Fredholm determinants and inverse scattering theory* A Brownian motion model of the matrices* Computation of the mean and of the variance of a number of statistical quantities* Selberg's integral and its consequences

Table of contents

• Front Cover
• Random Matrices: Revised and Enlarged
• Copyright Page
• Table of Contents
• Preface to the Second Edition
• Acknowledgments
• Preface to the First Edition
• Chapter 1. Introduction
• Chapter 2. Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements
• Chapter 3. Gaussian Ensembles. The Joint Probability Density Function for the Eigenvalues
• Chapter 4. Gaussian Ensembles. Level Density
• Chapter 5. Gaussian Unitary Ensemble
• Chapter 6. Gaussian Orthogonal Ensemble
• Chapter 7. Gaussian Symplectic Ensemble
• Chapter 8. Gaussian Ensembles: Brownian Motion Model
• Chapter 9. Circular Ensembles
• Chapter 10. Circular Ensembles (Continued)
• Chapter 11. Circular Ensembles. Thermodynamics
• Chapter 12. Asymptotic Behavior of Eβ( 0 , s) for Large s
• Chapter 13. Gaussian Ensemble of Antisymmetric Hermitian Matrices
• Chapter 14. Another Gaussian Ensemble of Hermitian Matrices
• Chapter 15. Matrices with Gaussian Element Densities but with No Unitary or Hermitian Conditions Imposed
• Chapter 16. Statistical Analysis of a Level Sequence
• Chapter 17. Selberg's Integral and Its Consequences
• Chapter 18. Gaussian Ensembles. Level Density in the Tail of the Semicircle
• Chapter 19. Restricted Trace Ensembles. Ensembles Related to the Classical Orthogonal Polynomials
• Chapter 20. Bordered Matrices
• Chapter 21. Invariance Hypothesis and Matrix Element Correlations
• Appendices
• Notes
• References
• Author Index
• Subject Index