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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.

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

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