Log-gases and random matrices / P.J. Forrester.

Author
Forrester, Peter (Peter John) [Browse]
Format
Book
Language
English
Published/​Created
Princeton : Princeton University Press, ©2010.
Description
xiv, 791 pages : illustrations ; 27 cm

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    Subject(s)
    Series
    • London Mathematical Society monographs ; new ser. no. 34. [More in this series]
    • The London Mathematical Society monographs series ; v. 34
    Summary note
    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.
    Bibliographic references
    Includes bibliographical references and index.
    Contents
    • Ch. 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 -ensemble
    • ch. 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 -ensemble
    • 2.9.Real orthogonal -ensemble
    • ch. 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 -ensemble
    • 3.11.Jacobi -ensemble
    • 3.12.Circular Jacobi -ensemble
    • ch. 4 The Selberg integral
    • 4.1.Selberg's derivation
    • 4.2.Anderson's derivation
    • 4.3.Consequences for the -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
    • ch. 5 Correlation functions at = 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
    • ch. 6 Correlation functions at = 1 and 4
    • 6.1.Correlation functions at = 4
    • 6.2.Construction of the skew orthogonal polynomials at = 4
    • 6.3.Correlation functions at = 1
    • 6.4.Construction of the skew orthogonal polynomials and summation formulas
    • 6.5.Alternate correlations at = 1
    • 6.6.Superimposed = 1 systems
    • 6.7.A two-component log-gas with charge ratio 1:2
    • ch. 7 Scaled limits at = 1, 2 and 4
    • 7.1.Scaled limits at = 2
    • Gaussian ensembles
    • 7.2.Scaled limits at = 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 = 2
    • 7.5.Bulk scaling and the zeros of the Riemann zeta function
    • 7.6.Scaled limits at = 4
    • Gaussian ensemble
    • 7.7.Scaled limits at = 4
    • 7.8.Scaled limits at = 1
    • 7.9.Scaled limits at = 1
    • 7.10.Two-component log-gas with charge ratio 1:2
    • ch. 8 Eigenvalue probabilities
    • Painleve systems approach
    • 8.1.Definitions
    • 8.2.Hamiltonian formulation of the Painleve theory
    • 8.3.-form Painleve equation characterizations
    • 8.4.The cases = 1 and 4
    • circular ensembles and bulk
    • 8.5.Discrete Painleve equations
    • 8.6.Orthogonal polynomial approach
    • ch. 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 Esoft (n; (s, )) for = 1, 4
    • 9.8.The probabilities Ehard (n; (O, s); a) for = 1, 4
    • 9.9.Riemann-Hilbert viewpoint
    • 9.10.Nonlinear equations from the Virasoro constraints
    • ch. 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
    • ch. 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 Schrodinger operators with exchange terms
    • 11.5.The operators H(H, Ex), H(L, Ex) and H(J, Ex)
    • 11.6.Dynamical correlations for = 2
    • 11.7.Scaled limits
    • ch. 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
    • ch. 13 Correlations for general
    • 13.1.Hypergeometric functions and Selberg correlation integrals
    • 13.2.Correlations at even
    • 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 ensemble
    • ch. 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(n; J) and P(n; J)
    • 14.7.Dynamical correlations
    • ch. 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.
    ISBN
    • 9780691128290 ((hardcover ; : alk. paper))
    • 0691128294 ((hardcover ; : alk. paper))
    LCCN
    2009053314
    OCLC
    466341422
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