Random Matrices

by Mehta, Madan Lal

• ISBN13: 9780124880511
• ISBN10: 0124880517
• Edition: 2nd
• Format: Hardcover
• Copyright: 1991-01-01
• Publisher: Academic Pr

Summary

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

Table of Contents

Gaussian Ensembles

The Joint Probability Density Function of the Matrix Elements

Gaussian Ensembles

The Joint Probability Density Function of the Eigenvalues

Gaussian Ensembles

Level Density

Gaussian Unitary Ensemble

Gaussian Orthogonal Ensemble

Gaussian Symplectic Ensemble

Brownian Motion Model

Circular Ensembles

Circular Ensembles (Continued)

Circular Ensembles

Thermodynamics

Asymptotic Behaviour of B(O,s) for Large S

Gaussian Ensemble of Anti-Symmetric Hermitian Matrices

Another Gaussian Ensemble of Hermitian Matrices

Matrices with Gaussian Element Densities but with No Unitary or Hermitian Condition Imposed

Statistical Analysis of a Level Sequence

Selberg's Integral and Its Consequences

Gaussian Ensembles

Level Density in the Tail of the Semi-Circle

Restricted Trace Ensembles

Bordered Matrices

Invariance Hypothesis and Matrix Element Correlations

Index

Table of Contents provided by Publisher. All Rights Reserved.