Random Matrices

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.

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.

@introbul:Key Features @bul:* 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

Physicists, mathematicians, and any scientist working in the fields of nuclear physics, solid state (especially amorphous material), chaotic systems, hydrodynamics, structural acoustics, multiple integrals, orthogonal polynomials, applied mathematical methods, zeta functions, and Dirichlet series. Prerequisite: Undergraduate mathematics, although subjects are treated at an advanced level.