# Random Matrices: Revised and Enlarged Second Edition

Elsevier, 19 ҐЁрт. 2014 Ё. - 562 ёҐюЁ.
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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

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 Chapter 1 Introduction 1 Chapter 2 Gaussian Ensembles The Joint Probability Density Function for the Matrix Elements 36 Chapter 3 Gaussian Ensembles The Joint Probability Density Function for the Eigenvalues 55 Chapter 4 Gaussian Ensembles Level Density 70 Chapter 5 Gaussian Unitary Ensemble 79 Chapter 6 Gaussian Orthogonal Ensemble 123 Chapter 7 Gaussian Symplectic Ensemble 162 Brownian Motion Model 170
 Chapter 15 Matrices with Gaussian Element Densities but with No Unitary or Hermitian Conditions Imposed 294 Chapter 16 Statistical Analysis of a Level Sequence 311 Chapter 17 Selbergs Integral and Its Consequences 339 Chapter 18 Gaussian Ensembles Level Density in the Tail of the Semicircle 371 Chapter 19 Restricted Trace Ensembles Ensembles Related to the Classical Orthogonal Polynomials 377 Chapter 20 Bordered Matrices 386 Chapter 21 Invariance Hypothesis and Matrix Element Correlations 394 Appendices 400

 Chapter 9 Circular Ensembles 181 Chapter 10 Circular Ensembles Continued 194 Chapter 11 Circular Ensembles Thermodynamics 224 Chapter 12 Asymptotic Behavior of Eβ 0 s for Large s 239 Chapter 13 Gaussian Ensemble of Antisymmetric Hermitian Matrices 260 Chapter 14 Another Gaussian Ensemble of Hermitian Matrices 267
 Notes 535 References 545 Author Index 555 Subject Index 559 └тҐюЁё№ъ│ яЁртр