## Random MatricesThis book 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. ˇ Presentation of many new results in one place for the first time. ˇ First time coverage of skew-orthogonal and bi-orthogonal polynomials and their use in the evaluation of some multiple integrals. ˇ Fredholm determinants and Painlevé equations. ˇ The three Gaussian ensembles (unitary, orthogonal, and symplectic); their n-point correlations, spacing probabilities. ˇ Fredholm determinants and inverse scattering theory. ˇ Probability densities of random determinants. |

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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 | |

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### Common terms and phrases

Appendix asymptotic average bi-orthogonal calculate Chapter cluster functions coefficient column complex computed const constant correlation and cluster correlation function corresponding defined derived diagonal elements differential equation dx dy Dyson eigenvalues expression finite Fredholm determinants Gaussian ensembles Gaussian orthogonal ensemble Gaussian unitary ensemble given by Eq gives Hermitian matrices independent integral equation integrand invariant joint probability density kernel Lemma level density linear matrix elements Mehta Meijer G-function Mellin transform monic polynomials multiple n x n n-level nuclear orthogonal polynomials Painlevé equation parameters permutations Pfaffian probability density function properties quaternion quaternion matrix random matrix real symmetric replace result Riemann zeta function rows satisfy Section self-dual skew-orthogonal solutions statistical symmetric matrix symplectic ensemble Theorem unitary matrix values variables Wigner write zero

### Popular passages

Page 4 - Hamiltonians is considered, each of which could describe a different nucleus. There is a strong logical expectation, though no rigorous mathematical proof, that an ensemble average will correctly describe the behaviour of one particular system which is under observation. The expectation is strong, because the system might be...

Page 4 - We wish to make statements about the fine detail of the level structure, and such statements cannot be made in terms of an ensemble of states. What is here required is a new kind of statistical mechanics, in which we renounce exact knowledge not of the state of a system but of the nature of the system itself. We picture a complex nucleus as a "black box" in which a large number of particles are interacting according to unknown laws.

Page 64 - The first term in W represents a harmonic potential which attracts each charge independently toward the point x = 0; the second term represents an electrostatic repulsion between each pair of charges. The logarithmic function comes in if we assume the universe to be two-dimensional. Let this charged gas be in thermodynamical equilibrium at a temperature T, so that the probability density of the positions of the TV charges is given by ., (4.2) where k is the Boltzmann constant.

Page 4 - The result of such an inquiry will be a statistical theory of energy levels. The statistical theory will not predict the detailed sequence of levels in any one nucleus, but it will describe the general appearance and the degree of irregularity of the level structure that is expected to occur in any nucleus which is too complicated to be understood in detail.

Page 42 - ... ensemble to be applicable, the splitting of the levels by the magnetic field must be at least as large as the average level spacing in the absence of the field. The magnetic interaction must in fact be so strong that it completely "mixes up" the level structure which would exist in zero field. Such a state of affairs could never occur in nuclear physics ; in atomic or molecular physics a practical application of the unitary ensemble may perhaps be possible. A system without invariance under time...

Page 54 - Thus the eigenvalues of H consist of N equal pairs. The Hamiltonian of any system which is invariant under time reversal and has odd spin satisfies the conditions of Theorem 2. All energy levels of such a system must be doubly degenerate. This is the Kramers degeneracy,14 and Theorem 2 shows how it appears naturally in the quaternion language. An immediate extension of Theorem 2 states that if Si and Si are two commuting...

Page 4 - ... ensemble average will correctly describe the behaviour of one particular system which is under observation. The expectation is strong, because the system might be one of a huge variety of systems, and very few of them will deviate much from a properly chosen ensemble average. On the other hand, our assumption that the ensemble average correctly describes a particular system, say the U239 nucleus, is not compelling. In fact, if this particular nucleus turns out to be far removed from the ensemble...