Random Matrices : Revised and Enlarged Second Edition
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 matrice
eBook, English, 2014
2nd ed
Elsevier Science, Saint Louis, 2014
1 online resource (581 pages).
9781483295954, 1483295958
1041746228
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; 1.1. Random Matrices in Nuclear Physics; 1.2. Random Matrices in Other Branches of Knowledge; 1.3. A Summary of Statistical Facts about Nuclear Energy Levels; 1.4. Definition of a Suitable Function for the Study of Level Correlations; 1.5. Wigner Surmise; 1.6. Electromagnetic Properties of Small Metallic Particles; 1.7. Analysis of Experimental Nuclear Levels; 1.8. The Zeros of the Riemann Zeta Function. 1.9. Things Worth Consideration, but Not Treated in This BookChapter 2. Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements; 2.1. Preliminaries; 2.2. Time-Reversal Invariance; 2.3. Gaussian Orthogonal Ensemble; 2.4. Gaussian Symplectic Ensemble; 2.5. Gaussian Unitary Ensemble; 2.6. Joint Probability Density Function for Matrix Elements; 2.7. Another Gaussian Ensemble of Hermitian Matrices; 2.8. Antisymmetric Hermitian Matrices; Summary of Chapter 2; Chapter 3. Gaussian Ensembles. The Joint Probability Density Function for the Eigenvalues. 3.1. Orthogonal Ensemble3.2. Symplectic Ensemble; 3.3. Unitary Ensemble; 3.4. Ensemble of Antisymmetric Hermitian Matrices; 3.5. Another Gaussian Ensemble of Hermitian Matrices; 3.6. Random Matrices and Information Theory; Summary of Chapter 3; Chapter 4. Gaussian Ensembles. Level Density; 4.1. The Partition Function; 4.2. The Asymptotic Formula for the Level Density. Gaussian Ensembles; 4-3. The Asymptotic Formula for the Level Density. Other Ensembles; Summary of Chapter 4; Chapter 5. Gaussian Unitary Ensemble; 5.1. Generalities; 5.2. The n-Point Correlation Function; 5.3. Level Spacings. 5.4. Several Consecutive Spacings5.5. Some Remarks; Summary of Chapter 5; Chapter 6. Gaussian Orthogonal Ensemble; 6.1. Generalities; 6.2. Quaternion Matrices; 6.3. The Probability Density Function as a Quaternion Determinant; 6.4. The Correlation and Cluster Functions; 6.5. Level Spacings. Integration over Alternate Variables; 6.6. Several Consecutive Spacings: n = 2r; 6.7. Several Consecutive Spacings: n = 2r
1; 6.8. Bounds for the Distribution Function of the Spacings; Summary of Chapter 6; Chapter 7. Gaussian Symplectic Ensemble; 7.1. A Quaternion Determinant. 7.2. Correlation and Cluster Functions7.3. Level Spacings; Summary of Chapter 7; Chapter 8. Gaussian Ensembles: Brownian Motion Model; 8.1. Stationary Ensembles; 8.2. Nonstationary Ensembles; 8.3. Some Ensemble Averages; Summary of Chapter 8; Chapter 9. Circular Ensembles; 9.1. The Orthogonal Ensemble; 9.2. Symplectic Ensemble; 9.3. Unitary Ensemble; 9.4. The Joint Probability Density Function for the Eigenvalues; Summary of Chapter 9; Chapter 10. Circular Ensembles (Continued); 10.1. Unitary Ensemble. Correlation and Cluster Functions; 10.2. Unitary Ensemble. Level Spacings
10.3. Orthogonal Ensemble. Correlation and Cluster Functions