Random Matrices: Revised and Enlarged Second Edition

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Elsevier, 19 трав. 2014 р. - 562 стор.
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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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Зміст

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

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