Cover -- Title -- Copyright -- Preface -- Contents -- Chapter 1. Gaussian matrix ensembles -- 1.1 Random real symmetric matrices -- 1.2 The eigenvalue p.d.f. for the GOE -- 1.3 Random complex Hermitian and quaternion real Hermitian matrices -- 1.4 Coulomb gas analogy -- 1.5 High-dimensional random energy landscapes -- 1.6 Matrix integrals and combinatorics -- 1.7 Convergence -- 1.8 The shifted mean Gaussian ensembles -- 1.9 Gaussian β-ensemble -- Chapter 2. Circular ensembles -- 2.1 Scattering matrices and Floquet operators -- 2.2 Definitions and basic properties -- 2.3 The elements of a random unitary matrix -- 2.4 Poisson kernel -- 2.5 Cauchy ensemble -- 2.6 Orthogonal and symplectic unitary random matrices -- 2.7 Log-gas systems with periodic boundary conditions -- 2.8 Circular β-ensemble -- 2.9 Real orthogonal β-ensemble -- Chapter 3. Laguerre and Jacobi ensembles -- 3.1 Chiral random matrices -- 3.2 Wishart matrices -- 3.3 Further examples of the Laguerre ensemble in quantum mechanics -- 3.4 The eigenvalue density -- 3.5 Correlated Wishart matrices -- 3.6 Jacobi ensemble and Wishart matrices -- 3.7 Jacobi ensemble and symmetric spaces -- 3.8 Jacobi ensemble and quantum conductance -- 3.9 A circular Jacobi ensemble -- 3.10 Laguerre β-ensemble -- 3.11 Jacobi β-ensemble -- 3.12 Circular Jacobi β-ensemble -- Chapter 4. The Selberg integral -- 4.1 Selberg's derivation -- 4.2 Anderson's derivation -- 4.3 Consequences for the β-ensembles -- 4.4 Generalization of the Dixon-Anderson integral -- 4.5 Dotsenko and Fateev's derivation -- 4.6 Aomoto's derivation -- 4.7 Normalization of the eigenvalue p.d.f.'s -- 4.8 Free energy -- Chapter 5. Correlation functions at β = 2 -- 5.1 Successive integrations -- 5.2 Functional differentiation and integral equation approaches -- 5.3 Ratios of characteristic polynomials -- 5.4 The classical weights.

5.5 Circular ensembles and the classical groups -- 5.6 Log-gas systems with periodic boundary conditions -- 5.7 Partition function in the case of a general potential -- 5.8 Biorthogonal structures -- 5.9 Determinantal k-component systems -- Chapter 6. Correlation functions at β = 1 and 4 -- 6.1 Correlation functions at β = 4 -- 6.2 Construction of the skew orthogonal polynomials at β = 4 -- 6.3 Correlation functions at β = 1 -- 6.4 Construction of the skew orthogonal polynomials and summation formulas -- 6.5 Alternate correlations at β = 1 -- 6.6 Superimposed β = 1 systems -- 6.7 A two-component log-gas with charge ratio 1:2 -- Chapter 7. Scaled limits at β = 1, 2 and 4 -- 7.1 Scaled limits at β = 2 - Gaussian ensembles -- 7.2 Scaled limits at β = 2 - Laguerre and Jacobi ensembles -- 7.3 Log-gas systems with periodic boundary conditions -- 7.4 Asymptotic behavior of the one- and two-point functions at β = 2 -- 7.5 Bulk scaling and the zeros of the Riemann zeta function -- 7.6 Scaled limits at β = 4 - Gaussian ensemble -- 7.7 Scaled limits at β = 4 - Laguerre and Jacobi ensembles -- 7.8 Scaled limits at β = 1 - Gaussian ensemble -- 7.9 Scaled limits at β = 1 - Laguerre and Jacobi ensembles -- 7.10 Two-component log-gas with charge ratio 1:2 -- Chapter 8. Eigenvalue probabilities - Painlevé systems approach -- 8.1 Definitions -- 8.2 Hamiltonian formulation of the Painlevé theory -- 8.3 σ-form Painlevé equation characterizations -- 8.4 The cases β = 1 and 4 - circular ensembles and bulk -- 8.5 Discrete Painlevé equations -- 8.6 Orthogonal polynomial approach -- Chapter 9. Eigenvalue probabilities - Fredholm determinant approach -- 9.1 Fredholm determinants -- 9.2 Numerical computations using Fredholm determinants -- 9.3 The sine kernel -- 9.4 The Airy kernel -- 9.5 Bessel kernels -- 9.6 Eigenvalue expansions for gap probabilities.

9.7 The probabilities E[sub(β)][sup(soft)] (n -- (s, ∞)) for β = 1, 4 -- 9.8 The probabilities E[sub(β)][sup(hard)] ( n -- (0, s) -- a) for β = 1, 4 -- 9.9 Riemann-Hilbert viewpoint -- 9.10 Nonlinear equations from the Virasoro constraints -- Chapter 10. Lattice paths and growth models -- 10.1 Counting formulas for directed nonintersecting paths -- 10.2 Dimers and tilings -- 10.3 Discrete polynuclear growth model -- 10.4 Further interpretations and variants of the RSK correspondence -- 10.5 Symmetrized growth models -- 10.6 The Hammersley process -- 10.7 Symmetrized permutation matrices -- 10.8 Gap probabilities and scaled limits -- 10.9 Hammersley process with sources on the boundary -- Chapter 11. The Calogero-Sutherland model -- 11.1 Shifted mean parameter-dependent Gaussian random matrices -- 11.2 Other parameter-dependent ensembles -- 11.3 The Calogero-Sutherland quantum systems -- 11.4 The Schrödinger operators with exchange terms -- 11.5 The operators H[sup((H, Ex))], H[sup((L, Ex))] and H[sup((J, Ex))] -- 11.6 Dynamical correlations for β = 2 -- 11.7 Scaled limits -- Chapter 12. Jack polynomials -- 12.1 Nonsymmetric Jack polynomials -- 12.2 Recurrence relations -- 12.3 Application of the recurrences -- 12.4 A generalized binomial theorem and an integration formula -- 12.5 Interpolation nonsymmetric Jack polynomials -- 12.6 The symmetric Jack polynomials -- 12.7 Interpolation symmetric Jack polynomials -- 12.8 Pieri formulas -- Chapter 13. Correlations for general β -- 13.1 Hypergeometric functions and Selberg correlation integrals -- 13.2 Correlations at even β -- 13.3 Generalized classical polynomials -- 13.4 Green functions and zonal polynomials -- 13.5 Inter-relations for spacing distributions -- 13.6 Stochastic differential equations -- 13.7 Dynamical correlations in the circular β ensemble.

Chapter 14. Fluctuation formulas and universal behavior of correlations -- 14.1 Perfect screening -- 14.2 Macroscopic balance and density -- 14.3 Variance of a linear statistic -- 14.4 Gaussian fluctuations of a linear statistic -- 14.5 Charge and potential fluctuations -- 14.6 Asymptotic properties of E[sub(β)](n -- J) and P[sub(β)](n -- J) -- 14.7 Dynamical correlations -- Chapter 15. The two-dimensional one-component plasma -- 15.1 Complex random matrices and polynomials -- 15.2 Quantum particles in a magnetic field -- 15.3 Correlation functions -- 15.4 General properties of the correlations and fluctuation formulas -- 15.5 Spacing distributions -- 15.6 The sphere -- 15.7 The pseudosphere -- 15.8 Metallic boundary conditions -- 15.9 Antimetallic boundary conditions -- 15.10 Eigenvalues of real random matrices -- 15.11 Classification of non-Hermitian random matrices -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.