Log-Gases and Random Matrices (LMS-34) (London Mathematical Society Monographs)

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, making Log-Gases and Random Matrices an indispensable reference work, as well as a learning resource for all students and researchers in the field. Title......Page 4 Copyright......Page 5 Preface......Page 6 Contents......Page 12 1.1 Random real symmetric matrices......Page 16 1.2 The eigenvalue p.d.f. for the GOE......Page 20 1.3 Random complex Hermitian and quaternion real Hermitian matrices......Page 26 1.4 Coulomb gas analogy......Page 35 1.5 High-dimensional random energy landscapes......Page 45 1.6 Matrix integrals and combinatorics......Page 48 1.7 Convergence......Page 56 1.8 The shifted mean Gaussian ensembles......Page 57 1.9 Gaussian β-ensemble......Page 58 2.1 Scattering matrices and Floquet operators......Page 68 2.2 Definitions and basic properties......Page 71 2.3 The elements of a random unitary matrix......Page 76 2.4 Poisson kernel......Page 81 2.5 Cauchy ensemble......Page 83 2.6 Orthogonal and symplectic unitary random matrices......Page 86 2.7 Log-gas systems with periodic boundary conditions......Page 88 2.8 Circular β-ensemble......Page 91 2.9 Real orthogonal β-ensemble......Page 96 3.1 Chiral random matrices......Page 100 3.2 Wishart matrices......Page 105 3.3 Further examples of the Laguerre ensemble in quantum mechanics......Page 113 3.4 The eigenvalue density......Page 121 3.5 Correlated Wishart matrices......Page 125 3.6 Jacobi ensemble and Wishart matrices......Page 126 3.7 Jacobi ensemble and symmetric spaces......Page 130 3.8 Jacobi ensemble and quantum conductance......Page 133 3.9 A circular Jacobi ensemble......Page 140 3.10 Laguerre β-ensemble......Page 142 3.11 Jacobi β-ensemble......Page 144 3.12 Circular Jacobi β-ensemble......Page 145 4.1 Selberg's derivation......Page 148 4.2 Anderson's derivation......Page 152 4.3 Consequences for the β-ensembles......Page 160 4.4 Generalization of the Dixon-Anderson integral......Page 171 4.5 Dotsenko and Fateev's derivation......Page 175 4.6 Aomoto's derivation......Page 180 4.7 Normalization of the eigenvalue p.d.f.'s......Page 187 4.8 Free energy......Page 195 5.1 Successive integrations......Page 201 5.2 Functional differentiation and integral equation approaches......Page 208 5.3 Ratios of characteristic polynomials......Page 212 5.4 The classical weights......Page 215 5.5 Circular ensembles and the classical groups......Page 222 5.6 Log-gas systems with periodic boundary conditions......Page 227 5.7 Partition function in the case of a general potential......Page 232 5.8 Biorthogonal structures......Page 238 5.9 Determinantal k-component systems......Page 244 6.1 Correlation functions at β = 4......Page 251 6.2 Construction of the skew orthogonal polynomials at β = 4......Page 261 6.3 Correlation functions at β = 1......Page 266 6.4 Construction of the skew orthogonal polynomials and summation formulas......Page 278 6.5 Alternate correlations at β = 1......Page 284 6.6 Superimposed β = 1 systems......Page 289 6.7 A two-component log-gas with charge ratio 1:2......Page 293 7.1 Scaled limits at β = 2 — Gaussian ensembles......Page 298 7.2 Scaled limits at β = 2 — Laguerre and Jacobi ensembles......Page 305 7.3 Log-gas systems with periodic boundary conditions......Page 312 7.4 Asymptotic behavior of the one- and two-point functions at β = 2......Page 313 7.5 Bulk scaling and the zeros of the Riemann zeta function......Page 316 7.6 Scaled limits at β = 4 — Gaussian ensemble......Page 323 7.7 Scaled limits at β = 4 — Laguerre and Jacobi ensembles......Page 327 7.8 Scaled limits at β = 1 — Gaussian ensemble......Page 331 7.9 Scaled limits at β = 1 — Laguerre and Jacobi ensembles......Page 334 7.10 Two-component log-gas with charge ratio 1:2......Page 338 8.1 Definitions......Page 343 8.2 Hamiltonian formulation of the Painlevé theory......Page 348 8.3 σ-form Painlevé equation characterizations......Page 364 8.4 The cases β = 1 and 4 — circular ensembles and bulk......Page 378 8.5 Discrete Painlevé equations......Page 387 8.6 Orthogonal polynomial approach......Page 390 9.1 Fredholm determinants......Page 395 9.2 Numerical computations using Fredholm determinants......Page 400 9.3 The sine kernel......Page 401 9.4 The Airy kernel......Page 408 9.5 Bessel kernels......Page 414 9.6 Eigenvalue expansions for gap probabilities......Page 418 9.7 The probabilities E[sub(β)][sup(soft)] (n; (s, ∞)) for β = 1, 4......Page 431 9.8 The probabilities E[sub(β)][sup(hard)] ( n; (0, s); a) for β = 1, 4......Page 436 9.9 Riemann-Hilbert viewpoint......Page 441 9.10 Nonlinear equations from the Virasoro constraints......Page 450 10.1 Counting formulas for directed nonintersecting paths......Page 455 10.2 Dimers and tilings......Page 471 10.3 Discrete polynuclear growth model......Page 478 10.4 Further interpretations and variants of the RSK correspondence......Page 486 10.5 Symmetrized growth models......Page 495 10.6 The Hammersley process......Page 502 10.7 Symmetrized permutation matrices......Page 507 10.8 Gap probabilities and scaled limits......Page 510 10.9 Hammersley process with sources on the boundary......Page 515 11.1 Shifted mean parameter-dependent Gaussian random matrices......Page 520 11.2 Other parameter-dependent ensembles......Page 527 11.3 The Calogero-Sutherland quantum systems......Page 531 11.4 The Schrödinger operators with exchange terms......Page 536 11.5 The operators H[sup((H, Ex))], H[sup((L, Ex))] and H[sup((J, Ex))]......Page 539 11.6 Dynamical correlations for β = 2......Page 545 11.7 Scaled limits......Page 555 12.1 Nonsymmetric Jack polynomials......Page 558 12.2 Recurrence relations......Page 565 12.3 Application of the recurrences......Page 568 12.4 A generalized binomial theorem and an integration formula......Page 570 12.5 Interpolation nonsymmetric Jack polynomials......Page 573 12.6 The symmetric Jack polynomials......Page 579 12.7 Interpolation symmetric Jack polynomials......Page 594 12.8 Pieri formulas......Page 598 13.1 Hypergeometric functions and Selberg correlation integrals......Page 607 13.2 Correlations at even β......Page 616 13.3 Generalized classical polynomials......Page 628 13.4 Green functions and zonal polynomials......Page 642 13.5 Inter-relations for spacing distributions......Page 648 13.6 Stochastic differential equations......Page 649 13.7 Dynamical correlations in the circular β ensemble......Page 655 14.1 Perfect screening......Page 673 14.2 Macroscopic balance and density......Page 678 14.3 Variance of a linear statistic......Page 680 14.4 Gaussian fluctuations of a linear statistic......Page 687 14.5 Charge and potential fluctuations......Page 695 14.6 Asymptotic properties of E[sub(β)](n; J) and P[sub(β)](n; J)......Page 703 14.7 Dynamical correlations......Page 713 15.1 Complex random matrices and polynomials......Page 716 15.2 Quantum particles in a magnetic field......Page 721 15.3 Correlation functions......Page 726 15.4 General properties of the correlations and fluctuation formulas......Page 733 15.5 Spacing distributions......Page 740 15.6 The sphere......Page 744 15.7 The pseudosphere......Page 753 15.8 Metallic boundary conditions......Page 759 15.9 Antimetallic boundary conditions......Page 762 15.10 Eigenvalues of real random matrices......Page 767 15.11 Classification of non-Hermitian random matrices......Page 775 Bibliography......Page 780 C......Page 800 E......Page 801 I......Page 802 M......Page 803 P......Page 804 S......Page 805 Z......Page 806 Random matrix theory, both as an application and as a theory, has evolved rapidly over the past 15 years. This book gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and jack polynomials