Preface to the Third Edition ................................. xiii
Preface to the Second Edition .................................. xv
Preface to the First Edition ................................. xvii
Chapter 1 Introduction ......................................... 1
1.1 Random Matrices in Nuclear Physics ......................... 1
1.2 Random Matrices in Other Branches of Knowledge ............. 5
1.3 A Summary of Statistical Facts about Nuclear Energy
Levels ..................................................... 8
1.3.1 Level Density ....................................... 8
1.3.2 Distribution of Neutron Widths ...................... 9
1.3.3 Radiation and Fission Widths ........................ 9
1.3.4 Level Spacings ..................................... 10
1.4 Def nition of a Suitable Function for the Study of Level
Correlations .............................................. 10
1.5 Wigner Surmise ............................................ 13
1.6 Electromagnetic Properties of Small Metallic Particles .... 15
1.7 Analysis of Experimental Nuclear Levels ................... 16
1.8 The Zeros of The Riemaqn Zeta Function .................... 16
1.9 Things Worth Consideration, But Not Treated in This Book .. 30
Chapter 2 Gaussian Ensembles. The Joint Probability Density
Function for the Matrix Elements ............................... 33
2.1 Preliminaries ............................................. 33
2.2 Time-Reversal Invariance .................................. 34
2.3 Gaussian Orthogonal Ensemble .............................. 36
2.4 Gaussian Symplectic Ensemble ............................. 38
2.5 Gaussian Unitary Ensemble ................................. 42
2.6 Joint Probability Density Function for the Matrix
Elements .................................................. 43
2.7 Gaussian Ensemble of Hermitian Matrices With Unequal
Real and Imaginary Parts .................................. 48
2.8 Anti-Symmetric Hermitian Matrices ......................... 48
Summary of Chapter 2 ........................................... 49
Chapter 3 Gaussian Ensembles. The Joint Probability Density
Function for the Eigenvalues ................................... 50
3.1 Orthogonal Ensemble ....................................... 50
3.2 Symplectic Ensemble ....................................... 54
3.3 Unitary Ensemble .......................................... 56
3.4 Ensemble of Anti-Symmetric Hermitian Matrices ............. 59
3.5 Gaussian Ensemble of Hermitian Matrices With Unequal
Real and Imaginary Parts .................................. 60
3.6 Random Matrices and Information Theory .................... 60
Summary of Chapter 3 ........................................... 62
Chapter 4 Gaussian Ensembles Level Density .................... 63
4.1 The Partition Function .................................... 63
4.2 The Asymptotic Formula for the Level Density. Gaussian
Ensembles ................................................. 65
4.3 The Asymptotic Formula for the Level Density. Other
Ensembles ................................................. 67
Summary of Chapter 4 ........................................... 69
Chapter 5 Orthogonal, Skew-Orthogonal and Bi-Orthogonal
Polynomials .................................................... 71
5.1 Quaternions, Pfaff ans, Determinants ...................... 72
5.2 Average Value of ΠNj = 1ƒ(xj); Orthogonal and
Skew-Orthogonal Polynomials ............................... 77
5.3 Case β = 2; Orthogonal Polynomials ........................ 78
5.4 Case β = 4; Skew-Orthogonal Polynomials of Quaternion
Type ...................................................... 82
5.5 Case β = 1; Skew-Orthogonal Polynomials of Real Type ...... 84
5.6 Average Value of ΠNj = 1ψ(xj, yj) Bi-Orthogonal
Polynomials ............................................... 88
5.7 Correlation Functions ..................................... 89
5.8 Proof of Theorem 5.7.1 .................................... 93
5.8.1 Case β = 2 ......................................... 93
5.8.2 Case β = 4 ......................................... 94
5.8.3 Case β = 1, Even Number of Variables ............... 96
5.8.4 Case β = 1, Odd Number of Variables ................ 99
5.9 Spacing Functions ........................................ 101
5.10 Determinanta! Representations ............................ 101
5.11 Integral Representations ................................. 103
5.12 Properties of the Zeros .................................. 106
5.13 Orthogonal Polynomials and the Riemann-Hilbert Problem ... 107
5.14 A Remark (Balian) ........................................ 108
Summary of Chapter 5 .......................................... 108
Chapter 6 Gaussian Unitary Ensemble .......................... 110
6.1 Generalities ............................................. 1ll
6.1.1 About Correlation and Cluster Functions ........... 111
6.1.2 About Level-Spacings .............................. 113
6.1.3 Spacing Distribution .............................. 118
6.1.4 Correlations and Spacings ......................... 118
6.2 The n-Point Correlation Function ......................... 118
6.3 Level Spacings ........................................... 122
6.4 Several Consecutive Spacings ............................. 127
6.5 Some Remarks ............................................. 134
Summary of Chapter 6 .......................................... 144
Chapter 7 Gaussian Orthogonal Ensemble ....................... 146
7.1 Generalities ............................................. 147
7.2 Correlation and Cluster Functions ........................ 148
7.3 Level Spacings. Integration Over Alternate Variables ..... 154
7.4 Several Consecutive Spacings: n = 2r ..................... 157
7.5 Several Consecutive Spacings: n = 2r — 1 ................. 162
7.5.1 Case n = 1 ........................................ 163
7.5.2 Case n = 2r - 1 ................................... 164
7.6 Bounds for the Distribution Function of the Spacings ..... 168
Summary of Chapter 7 .......................................... 172
Chapter 8 Gaussian Symplectic Ensemble ....................... 175
8.1 A Quaternion Determinant ................................. 175
8.2 Correlation and Cluster Functions ........................ 177
8.3 Level Spacings ........................................... 179
Summary of Chapter 8 .......................................... 181
Chapter 9 Gaussian Ensembles: Brownian Motion Model .......... 182
9.1 Stationary Ensembles ..................................... 182
9.2 Nonstationary Ensembles .................................. 183
9.3 Some Ensemble Averages ................................... 187
Summary of Chapter 9 .......................................... 189
Chapter 10 Circular Ensembles ................................. 191
10.1 Orthogonal Ensemble ...................................... 192
10.2 Symplectic Ensemble ...................................... 194
10.3 Unitary Ensemble ......................................... 196
10.4 The Joint Probability Density of the Eigenvalues ......... 197
Summary of Chapter 10 ......................................... 201
Chapter 11 Circular Ensembles (Continued) ..................... 203
11.1 Unitary Ensemble. Correlation and Cluster Functions ...... 203
11.2 Unitary Ensemble. Level Spacings ......................... 205
11.3 Orthogonal Ensemble. Correlation and Cluster
Functions ................................................ 207
11.3.1 The Case N = 2m, Even ............................. 209
11.3.2 The Case N = 2m + 1, Odd .......................... 210
11.3.3 Conditions of Theorem 5.1.4 ...................... 211
11.3.4 Correlation and Cluster Functions ................. 212
11.4 Orthogonal Ensemble. Level Spacings ...................... 213
11.5 Symplectic Ensemble. Correlation and Cluster Functions ... 216
11.6 Relation Between Orthogonal and Symplectic Ensembles ..... 218
11.7 Symplectic Ensemble. Level Spacings ...................... 219
11.8 Brownian Motion Model .................................... 221
Summary of Chapter 11 ......................................... 223
Chapter 12 Circular Ensembles. Thermodynamics ................. 224
12.1 The Partition Function ................................... 224
12.2 Thermodynamic Quantities ................................. 227
12.3 Statistical Interpretation of U and С .................... 229
12.4 Continuum Model for the Spacing Distribution ............. 231
Summary of Chapter 12 ......................................... 236
Chapter 13 Gaussian Ensemble of Anti-Symmetric Hermitian
Matrices ...................................................... 237
13.1 Level Density. Correlation Functions ..................... 237
13.2 Level Spacings ........................................... 240
13.2.1 Central Spacings .................................. 240
13.2.2 Non-Central Spacings .............................. 242
Summary of Chapter 13 ......................................... 243
Chapter 14 A Gaussian Ensemble of Hermitian Matrices With
Unequal Real and Imaginary Parts .............................. 244
14.1 Summary of Results. Matrix Ensembles From GOE to GUE
and Beyond ............................................... 245
14.2 Matrix Ensembles From GSE to GUE and Beyond .............. 250
14.3 Joint Probability Density for the Eigenvalues ............ 254
14.3.1 Matrices From GOE to GUE and Beyond ............... 256
14.3.2 Matrices From GSE to GUE and Beyond ............... 260
14.4 Correlation and Cluster Functions ........................ 263
Summary of Chapter 14 ......................................... 264
Chapter 15 Matrices With Gaussian Element Densities But
With No Unitary or Hermitian Conditions Imposed ............... 266
15.1 Complex Matrices ......................................... 266
15.2 Quaternion Matrices ...................................... 273
15.3 Real Matrices ............................................ 279
15.4 Determinants: Probability Densities ...................... 281
Summary of Chapter 15 ......................................... 286
Chapter 16 Statistical Analysis of a Level-Sequence ........... 287
16.1 Linear Statistic or the Number Variance .................. 290
16.2 Least Square Statistic ................................... 294
16.3 Energy Statistic ......................................... 298
16.4 Covariance of Two Consecutive Spacings ................... 301
16.5 The F-Statistic .......................................... 302
16.6 The Λ-Statistic .......................................... 303
16.7 Statistics Involving Three and Four Level Correlations ... 303
16.8 Other Statistics ......................................... 307
Summary of Chapter 16 ......................................... 308
Chapter 17 Selberg's Integral and Its Consequences ............ 309
17.1 Selberg's Integral ....................................... 309
17.2 Selberg's Proof of Eq. (17.1.3) .......................... 311
17.3 Aomoto's Proof of Eqs. (17.1.4) and (17.1.3) ............. 315
17.4 Other Averages ........................................... 318
17.5 Other Forms of Selberg's Integral ........................ 318
17.6 Some Consequences of Selberg's Integral .................. 320
17.7 Normalization Constant for the Circular Ensembles ........ 323
17.8 Averages With Laguerre or Hermite Weights ................ 323
17.9 Connection With Finite Reflectio Groups .................. 325
17.10 A Second Generalization of the Beta Integral ............ 327
17.11 Some Related Diff cult Integrals ........................ 329
Summary to Chapter 17 ......................................... 334
Chapter 18 Asymptotic Behaviour of Eβ(0, s) by Inverse
Scattering .................................................... 335
18.1 Asymptotics of λn(t) ..................................... 336
18.2 Asymptotics of Toeplitz Determinants ..................... 339
18.3 Fredholm Determinants and the Inverse Scattering Theory .. 340
18.4 Application of the Gel'fаnd-Levitan Method ............... 342
18.5 Application of the Marchenko Method ...................... 347
18.6 Asymptotic Expansions .................................... 350
Summary of Chapter 18 ......................................... 353
Chapter 19 Matrix Ensembles and Classical Orthogonal
Polynomials ................................................... 354
19.1 Unitary Ensemble ......................................... 355
19.2 Orthogonal Ensemble ...................................... 357
19.3 Symplectic Ensemble ...................................... 361
19.4 Ensembles With Other Weights ............................. 363
19.5 Conclusion ............................................... 363
Summary of Chapter 19 ......................................... 364
Chapter 20 Level Spacing Functions Eβ(r,s); Inter-relations
and Power Series Expansions ................................... 365
20.1 Three Sets of Spacing Functions; Their Inter-Relations ... 365
20.2 Relation Between Odd and Even Solutions of Eq.
(20.1.13)................................................. 368
20.3 Relation Between F1(z, s) and F±(z,s) .................... 371
20.4 Relation Between F4(z,s) and F±(z,s) ..................... 375
20.5 Power Series Expansions of Eβ(r,s) ....................... 376
Summary of Chapter 20 ......................................... 381
Chapter 21 Fredholm Determinants and Painlevй Equations ....... 382
21.1 Introduction ............................................. 382
21.2 Proof of Eqs. (21.1.11)-(21.1.17) ........................ 385
21.3 Differential Equations for the Functions А, В and S ...... 394
21.4 Asymptotic Expansions for Large Positive r ............... 396
21.5 Fifth and Third Painlevé Transcendents ................... 400
21.6 Solution of Eq. (21.3.6) for Large t ..................... 406
Summary of Chapter 21 ......................................... 408
Chapter 22 Moments of the Characteristic Polynomial in the
Three Ensembles of Random Matrices ............................ 409
22.1 Introduction ............................................. 409
22.2 Calculation of Iβ(n,m;x) ................................. 411
22.2.1 Iβ(n,m;x) as a determinant or a Pfeffanofa matrix
ofsize depending on n ............................. 412
22.2.2 Iβ(n, m; x) as determinants of size
depending on m .................................... 415
22.3 Special Case of the Gaussian Weight ...................... 419
22.4 Average Value of Πmi = 1 det(xj1-А)Πℓi = 1 det(zj1-А)-1 ......... 421
Summary of Chapter 22 ......................................... 424
Chapter 23 Hermitian Matrices Coupled in a Chain .............. 426
23.1 General Correlation Function ............................. 428
23.2 Proof of Theorem 23.1.1 .................................. 430
23.3 Spacing Functions ........................................ 435
23.4 The Generating Function R(z1, I1; ... zp, Ip) ............ 437
23.5 The Zeros of the Bi-Orthogonal Polynomials ............... 441
Summary of Chapter 23 ......................................... 448
Chapter 24 Gaussian Ensembles. Edge of the Spectrum ........... 449
24.1 Level Density Near the Inf ection Point .................. 450
24.2 Spacing Functions ........................................ 452
24.3 Differential Equations; Painlevé ......................... 454
Summary to Chapter 24 ......................................... 458
Chapter 25 Random Permutations, Circular Unitary Ensemble
(CUE) and Gaussian Unitary Ensemble (GUE) ..................... 460
25.1 Longest Increasing Subsequences in Random Permutations ... 460
25.2 Random Permutations and the Circular Unitary Ensemble .... 461
25.3 Robinson-Schensted Correspondence ........................ 463
25.4 Random Permutations and GUE .............................. 468
Summary of Chapter 25 ......................................... 468
Chapter 26 Probability Densities of the Determinants;
Gaussian Ensembles ............................................ 469
26.1 Introduction ............................................. 469
26.2 Gaussian Unitary Ensemble ................................ 473
26.2.1 Mellin Transform of the PDD ....................... 473
26.2.2 Inverse Mellin Transforms ......................... 475
26.3 Gaussian Symplectic Ensemble ............................. 477
26.4 Gaussian Orthogonal Ensemble ............................. 480
26.5 Gaussian Orthogonal Ensemble. Case n = 2m + 1 Odd ........ 482
26.6 Gaussian Orthogonal Ensemble. Case n = 2m Even ........... 483
Summary of Chapter 26 ......................................... 486
Chapter 27 Restricted Trace Ensembles ......................... 487
27.1 Fixed Trace Ensemble; Equivalence of Moments ............. 487
27.2 Probability Density of the Determinant ................... 490
27.3 Bounded Trace Ensembles .................................. 492
Summary of Chapter 27 ......................................... 493
Appendices .................................................... 494
Notes ......................................................... 645
References .................................................... 655
Author Index .................................................. 680
Subject Index ................................................. 684
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