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Log-gases and random matrices / P.J. Forrester.
Author
Forrester, Peter (Peter John)
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Format
Book
Language
English
Published/Created
Princeton : Princeton University Press, ©2010.
Description
xiv, 791 pages : illustrations ; 27 cm
Availability
Available Online
Ebook Central Perpetual, DDA and Subscription Titles
De Gruyter Princeton University Press eBook-Package Backlist 2000-2013
De Gruyter Princeton University Press eBook Package 2000-2013
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Call Number
Status
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Notes
Lewis Library - Stacks
QA188 .F656 2010
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Subject(s)
Random matrices
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Jacobi polynomials
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Integral theorems
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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.
Show 147 more Contents items
ISBN
9780691128290 ((hardcover ; : alk. paper))
0691128294 ((hardcover ; : alk. paper))
LCCN
2009053314
OCLC
466341422
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Log-gases and random matrices [electronic resource] / P.J. Forrester.
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99125124880306421
Log-gases and random matrices / P.J. Forrester.
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SCSB-9104225