Random MatricesRandom Matrices gives a coherent and detailed description of analytical methods devised to study random matrices. These methods are critical to the understanding of various fields in in mathematics and mathematical physics, such as nuclear excitations, ultrasonic resonances of structural materials, chaotic systems, the zeros of the Riemann and other zeta functions. More generally they apply to the characteristic energies of any sufficiently complicated system and which have found, since the publication of the second edition, many new applications in active research areas such as quantum gravity, traffic and communications networks or stock movement in the financial markets. This revised and enlarged third edition reflects the latest developements in the field and convey a greater experience with results previously formulated. For example, the theory of skew-orthogoanl and bi-orthogonal polynomials, parallel to that of the widely known and used orthogonal polynomials, is explained here for the first time.
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Contents
1 | |
33 | |
50 | |
63 | |
71 | |
Chapter 6 Gaussian Unitary Ensemble | 110 |
Chapter 7 Gaussian Orthogonal Ensemble | 146 |
Chapter 8 Gaussian Symplectic Ensemble | 175 |
Chapter 18 Asymptotic Behaviour of Eβ 0 s by Inverse Scattering | 335 |
Chapter 19 Matrix Ensembles and Classical Orthogonal Polynomials | 354 |
Chapter 20 Level Spacing Functions Eβr s Interrelations and Power Series Expansions | 365 |
Chapter 21 Fredholm Determinants and Painlev Equations | 382 |
Chapter 22 Moments of the Characteristic Polynomial in the Three Ensembles of Random Matrices | 409 |
Chapter 23 Hermitian Matrices Coupled in a Chain | 426 |
Chapter 24 Gaussian Ensembles Edge of the Spectrum | 449 |
Chapter 25 Random Permutations Circular Unitary Ensemble CUE and Gaussian Unitary Ensemble GUE | 460 |
Brownian Motion Model | 182 |
Chapter 10 Circular Ensembles | 191 |
Chapter 11 Circular Ensembles Continued | 203 |
Chapter 12 Circular Ensembles Thermodynamics | 224 |
Chapter 13 Gaussian Ensemble of AntiSymmetric Hermitian Matrices | 237 |
Chapter 14 A Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts | 244 |
Chapter 15 Matrices With Gaussian Element Densities But With No Unitary or Hermitian Conditions Imposed | 266 |
Chapter 16 Statistical Analysis of a LevelSequence | 287 |
Chapter 17 Selbergs Integral and Its Consequences | 309 |
Chapter 26 Probability Densities of the Determinants Gaussian Ensembles | 469 |
Chapter 27 Restricted Trace Ensembles | 487 |
Appendices | 494 |
Notes | 645 |
655 | |
680 | |
684 | |