Lal Mehta Random Matrices

Revised and Enlarged Second Edition
2. Auflage 2014
ISBN: 978-1-4832-9595-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 562 Seiten, Web PDF

Reihe: Pure and Applied Mathematics

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

Autoren/Hrsg.

Lal Mehta, Madan

Inhaltsverzeichnis

1;Front Cover;1
2;Random Matrices: Revised and Enlarged;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface to the Second Edition;12
6;Acknowledgments;16
7;Preface to the First Edition;18
8;Chapter 1. Introduction;20
8.1;1.1. Random Matrices in Nuclear Physics;20
8.2;1.2. Random Matrices in Other Branches of Knowledge;25
8.3;1.3. A Summary of Statistical Facts about Nuclear Energy Levels;29
8.4;1.4. Definition of a Suitable Function for the Study of Level Correlations;33
8.5;1.5. Wigner Surmise;34
8.6;1.6. Electromagnetic Properties of Small Metallic Particles;37
8.7;1.7. Analysis of Experimental Nuclear Levels;39
8.8;1.8. The Zeros of the Riemann Zeta Function;40
8.9;1.9. Things Worth Consideration, but Not Treated in This Book;52
9;Chapter 2. Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements;55
9.1;2.1. Preliminaries;55
9.2;2.2. Time-Reversal Invariance;56
9.3;2.3. Gaussian Orthogonal Ensemble;58
9.4;2.4. Gaussian Symplectic Ensemble;60
9.5;2.5. Gaussian Unitary Ensemble;65
9.6;2.6. Joint Probability Density Function for Matrix Elements;66
9.7;2.7. Another Gaussian Ensemble of Hermitian Matrices;71
9.8;2.8. Antisymmetric Hermitian Matrices;72
9.9;Summary of Chapter 2;72
10;Chapter 3. Gaussian Ensembles. The Joint Probability Density Function for the Eigenvalues;74
10.1;3.1. Orthogonal Ensemble;74
10.2;3.2. Symplectic Ensemble;78
10.3;3.3. Unitary Ensemble;81
10.4;3.4. Ensemble of Antisymmetric Hermitian Matrices;84
10.5;3.5. Another Gaussian Ensemble of Hermitian Matrices;85
10.6;3.6. Random Matrices and Information Theory;86
10.7;Summary of Chapter 3;88
11;Chapter 4. Gaussian Ensembles. Level Density;89
11.1;4.1. The Partition Function;89
11.2;4.2. The Asymptotic Formula for the Level Density. Gaussian Ensembles;91
11.3;4-3. The Asymptotic Formula for the Level Density. Other Ensembles;94
11.4;Summary of Chapter 4;97
12;Chapter 5. Gaussian Unitary Ensemble;98
12.1;5.1. Generalities;99
12.2;5.2. The n-Point Correlation Function;108
12.3;5.3. Level Spacings;114
12.4;5.4. Several Consecutive Spacings;120
12.5;5.5. Some Remarks;128
12.6;Summary of Chapter 5;140
13;Chapter 6. Gaussian Orthogonal Ensemble;142
13.1;6.1. Generalities;142
13.2;6.2. Quaternion Matrices;144
13.3;6.3. The Probability Density Function as a Quaternion Determinant;147
13.4;6.4. The Correlation and Cluster Functions;154
13.5;6.5. Level Spacings. Integration over Alternate Variables;157
13.6;6.6. Several Consecutive Spacings: n = 2r;161
13.7;6.7. Several Consecutive Spacings: n = 2r – 1;166
13.8;6.8. Bounds for the Distribution Function of the Spacings;171
13.9;Summary of Chapter 6;179
14;Chapter 7. Gaussian Symplectic Ensemble;181
14.1;7.1. A Quaternion Determinant;181
14.2;7.2. Correlation and Cluster Functions;184
14.3;7.3. Level Spacings;186
14.4;Summary of Chapter 7;188
15;Chapter 8. Gaussian Ensembles: Brownian Motion Model;189
15.1;8.1. Stationary Ensembles;189
15.2;8.2. Nonstationary Ensembles;189
15.3;8.3. Some Ensemble Averages;195
15.4;Summary of Chapter 8;198
16;Chapter 9. Circular Ensembles;200
16.1;9.1. The Orthogonal Ensemble;201
16.2;9.2. Symplectic Ensemble;204
16.3;9.3. Unitary Ensemble;206
16.4;9.4. The Joint Probability Density Function for the Eigenvalues;207
16.5;Summary of Chapter 9;212
17;Chapter 10. Circular Ensembles (Continued);213
17.1;10.1. Unitary Ensemble. Correlation and Cluster Functions;213
17.2;10.2. Unitary Ensemble. Level Spacings;216
17.3;10.3. Orthogonal Ensemble. Correlation and Cluster Functions;218
17.4;10.4. Orthogonal Ensemble. Level Spacings;225
17.5;10.5. Symplectic Ensemble. Correlation and Cluster Functions;229
17.6;10.6. Relation between Orthogonal and Symplectic Ensembles;231
17.7;10.7. Symplectic Ensemble. Level Spacings;233
17.8;10.8. Brownian Motion Model;235
17.9;10.9. Wigner's Method for the Orthogonal Circular Ensemble;237
17.10;Summary of Chapter 10;241
18;Chapter 11. Circular Ensembles. Thermodynamics;243
18.1;11.1. The Partition Function;243
18.2;11.2. Thermodynamic Quantities;246
18.3;11.3. Statistical Interpretation of U and C;249
18.4;11.4. Continuum Model for the Spacing Distribution;251
18.5;Summary of Chapter 11;257
19;Chapter 12. Asymptotic Behavior of Eß( 0 , s) for Large s;258
19.1;12.1. Asymptotics of the ..( t );259
19.2;12.2. Asymptotics of Toeplitz Determinants;262
19.3;12.3. Fredholm Determinants and the Inverse Scattering Theory;263
19.4;12.4. Application of the Gel'fand–Levitan Method;266
19.5;12.5. Application of the Marchenko Method;271
19.6;12.6. Asymptotic Expansions;274
19.7;Summary of Chapter 12;277
20;Chapter 13. Gaussian Ensemble of Antisymmetric Hermitian Matrices;279
20.1;13.1. Level Density. Correlation Functions;279
20.2;13.2. Level Spacings;282
20.3;Summary of Chapter 13;285
21;Chapter 14. Another Gaussian Ensemble of Hermitian Matrices;286
21.1;14.1. Summary of Results. Matrix Ensembles from GOE to GUE and Beyond;287
21.2;14.2. Matrix Ensembles from GSE to GUE and Beyond;294
21.3;14.3. Joint Probability Density for the Eigenvalues;298
21.4;14.4. Correlation and Cluster Functions;309
21.5;Summary of Chapter 14;312
22;Chapter 15. Matrices with Gaussian Element Densities but with No Unitary or Hermitian Conditions Imposed;313
22.1;15.1. Complex Matrices;313
22.2;15.2. Quaternion Matrices;320
22.3;15.3. Real Matrices;328
22.4;Summary of Chapter 15;329
23;Chapter 16. Statistical Analysis of a Level Sequence;330
23.1;16.1. Linear Statistics or the Number Variance;333
23.2;16.2. Least Square Statistic;338
23.3;16.3. Energy Statistic;343
23.4;16.4. Covariance of Two Consecutive Spacings;346
23.5;16.5. The F Statistic;349
23.6;16.6. The A Statistic;351
23.7;16.7. Statistics Involving Three- and Four-Level Correlations;352
23.8;16.8. Other Statistics;353
23.9;Summary of Chapter 16;357
24;Chapter 17. Selberg's Integral and Its Consequences;358
24.1;17.1. Selberg's Integral;358
24.2;17.2. Selberg's Proof of Equation (17.1.3);359
24.3;17.3. Aomoto's Proof of Equation (17.1.4);364
24.4;17.4. Other Averages;368
24.5;17.5. Other Forms of Selberg's Integral;368
24.6;17.6. Some Consequences of Selberg's Integral;371
24.7;17.7. Normalization Constant for the Circular Ensembles;375
24.8;17.8. Averages with Laguerre or Hermite Weights;375
24.9;17.9. Connection with Finite Reflection Groups;378
24.10;17.10. A Second Generalization of the Beta Integral;380
24.11;17.11. Some Related Difficult Integrals;383
24.12;Summary of Chapter 17;388
25;Chapter 18. Gaussian Ensembles. Level Density in the Tail of the Semicircle;390
25.1;18.1. Level Density near the Inflection Point;391
25.2;Summary of Chapter 18;395
26;Chapter 19. Restricted Trace Ensembles. Ensembles Related to the Classical Orthogonal Polynomials;396
26.1;19.1. Fixed Trace Ensemble;396
26.2;19.2. Bounded Trace Ensemble;400
26.3;19.3. Matrix Ensembles and Classical Orthogonal Polynomials;401
26.4;Summary of Chapter 19;403
27;Chapter 20. Bordered Matrices;405
27.1;20.1. Random Linear Chain;406
27.2;20.2. Bordered Matrices;410
27.3;Summary of Chapter 20;412
28;Chapter 21. Invariance Hypothesis and Matrix Element Correlations;413
28.1;21.1. Random Orthonormal Vectors;414
28.2;Summary of Chapter 21;418
29;Appendices;419
30;Notes;554
31;References;564
32;Author Index;574
33;Subject Index;578