Log-gases and random matrices

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 ensembl...

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Hlavní autor:	Forrester, P.J
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	Princeton Princeton University Press 2010
Vydání:	1
Edice:	London Mathematical Society Monographs
Témata:	Asymptotic formula Bidiagonal matrix Binomial coefficient Calculation Central limit theorem Change of variables Characteristic polynomial Coefficient Correlation function Correlation function (quantum field theory) Cumulative distribution function Determinant Diagram (category theory) Differential equation Dirac equation Dirac operator Eigenfunction Eigenvalues and eigenvectors Elliptic function Equation Euler's formula Explicit formulae (L-function) Factorization Fredholm determinant Gamma function Geometric distribution Haar measure Hecke algebra Hermite polynomials Hermitian matrix Hessenberg matrix Integer matrix Integral equation Integral theorems Jacobi polynomials Joint probability distribution Laplace's equation Lattice path MATHEMATICS MATHEMATICS / Matrices MATHEMATICS / Probability & Statistics / General Nucleation Operator (physics) Orthogonal polynomials Partial differential equation Partial fraction decomposition Partition function (mathematics) Partition function (statistical mechanics) Permutation Permutation matrix Pfaffian Poisson point process Polynomial Power series Probability Probability density function Probability measure Quaternion Random matrices Random matrix Riemann sum Riemann zeta function Scaling limit Scatter matrix Schur polynomial Statistic Statistical mechanics Statistics Subsequence Symmetric matrix Symmetrization Symplectic matrix Trace formula Tridiagonal matrix Unitary matrix Volume form Polynomial Random matrix Nucleation Eigenfunction Pfaffian Trace formula Hermitian matrix Laplace's equation Lattice path Gamma function Partial differential equation Cumulative distribution function Euler's formula Eigenvalues and eigenvectors Probability density function Correlation function Calculation Integral equation Symplectic matrix Explicit formulae (L-function) Jacobi polynomials Bidiagonal matrix Volume form Permutation Determinant Elliptic function Geometric distribution Fredholm determinant Poisson point process Haar measure Partition function (mathematics) Partial fraction decomposition Partition function (statistical mechanics) Subsequence Hessenberg matrix Joint probability distribution Statistical mechanics Tridiagonal matrix Riemann sum Probability measure Symmetrization Quaternion Differential equation Scaling limit Change of variables Characteristic polynomial Operator (physics) Diagram (category theory) Schur polynomial Power series Asymptotic formula Hecke algebra Statistic Orthogonal polynomials Riemann zeta function Binomial coefficient Hermite polynomials Dirac operator Unitary matrix Coefficient Scatter matrix Probability Equation Factorization Integer matrix Correlation function (quantum field theory) Permutation matrix Central limit theorem Symmetric matrix Dirac equation
ISBN:	9781400835416, 1400835410, 9780691128290, 0691128294
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Log-gases and random matrices -- Preface -- Contents -- Chapter One: Gaussian Matrix Ensembles -- Chapter Two: Circular Ensembles -- Chapter Three: Laguerre and Jacobi Ensembles -- Chapter Four: The Selberg Integral -- Chapter Five: Correlation Functions at β = 2 -- Chapter Six: Correlation Functions at β = 1 and 4 -- Chapter Seven: Scaled Limits at β = 1, 2 and 4 -- Chapter Eight: Eigenvalue Probabilities-Painlevé Systems Approach -- Chapter Nine: Eigenvalue Probabilities-Fredholm Determinant Approach -- Chapter Ten: Lattice Paths and Growth Models -- Chapter Eleven: The Calogero-Sutherland Model -- Chapter Twelve: Jack polynomials -- Chapter Thirteen: Correlations for General β -- Chapter Fourteen: Fluctuation Formulas and Universal Behavior Of Correlations -- Chapter Fifteen: The Two-Dimensional One-Component Plasma -- Bibliography -- Index
Front Matter Preface Table of Contents Chapter One: Gaussian matrix ensembles Chapter Two: Circular ensembles Chapter Three: Laguerre and Jacobi ensembles Chapter Four: The Selberg integral Chapter Five: Correlation functions at $\beta = 2$ Chapter Six: Correlation functions at $\beta = 1$ and 4 Chapter Seven: Scaled limits at $\beta = 1,\;2$ and 4 Chapter Eight: Eigenvalue probabilities — Painlevé systems approach Chapter Nine: Eigenvalue probabilities — Fredholm determinant approach Chapter Ten: Lattice paths and growth models Chapter Eleven: The Calogero—Sutherland model Chapter Twelve: Jack polynomials Chapter Thirteen: Correlations for general $\beta $ Chapter Fourteen: Fluctuation formulas and universal behavior of correlations Chapter Fifteen: The two-dimensional one-component plasma Bibliography Index
12.8 Pieri formulas -- Chapter 13. Correlations for general &amp -- #946 -- -- 13.1 Hypergeometric functions and Selberg correlation integrals -- 13.2 Correlations at even &amp -- #946 -- -- 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 &amp -- #946 -- 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(&amp -- #946 -- )](n -- J) and P[sub(&amp -- #946 -- )](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
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(&amp -- #946 -- )][sup(soft)] (n -- (s, &amp -- #8734 -- )) for &amp -- #946 -- = 1, 4 -- 9.8 The probabilities E[sub(&amp -- #946 -- )][sup(hard)] ( n -- (0, s) -- a) for &amp -- #946 -- = 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 &amp -- #946 -- = 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
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 &amp -- #946 -- -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 &amp -- #946 -- -ensemble -- 2.9 Real orthogonal &amp -- #946 -- -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 &amp -- #946 -- -ensemble -- 3.11 Jacobi &amp -- #946 -- -ensemble -- 3.12 Circular Jacobi &amp -- #946 -- -ensemble -- Chapter 4. The Selberg integral -- 4.1 Selberg's derivation -- 4.2 Anderson's derivation -- 4.3 Consequences for the &amp -- #946 -- -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 &amp -- #946 -- = 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 &amp -- #946 -- = 1 and 4 -- 6.1 Correlation functions at &amp -- #946 -- = 4 -- 6.2 Construction of the skew orthogonal polynomials at &amp -- #946 -- = 4 -- 6.3 Correlation functions at &amp -- #946 -- = 1 -- 6.4 Construction of the skew orthogonal polynomials and summation formulas -- 6.5 Alternate correlations at &amp -- #946 -- = 1 -- 6.6 Superimposed &amp -- #946 -- = 1 systems -- 6.7 A two-component log-gas with charge ratio 1:2 -- Chapter 7. Scaled limits at &amp -- #946 -- = 1, 2 and 4 -- 7.1 Scaled limits at &amp -- #946 -- = 2 - Gaussian ensembles -- 7.2 Scaled limits at &amp -- #946 -- = 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 &amp -- #946 -- = 2 -- 7.5 Bulk scaling and the zeros of the Riemann zeta function -- 7.6 Scaled limits at &amp -- #946 -- = 4 - Gaussian ensemble -- 7.7 Scaled limits at &amp -- #946 -- = 4 - Laguerre and Jacobi ensembles -- 7.8 Scaled limits at &amp -- #946 -- = 1 - Gaussian ensemble -- 7.9 Scaled limits at &amp -- #946 -- = 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 &amp -- #963 -- -form Painlevé equation characterizations -- 8.4 The cases &amp -- #946 -- = 1 and 4 - circular ensembles and bulk
Chapter Twelve. Jack polynomials
Chapter Thirteen. Correlations for general β
Chapter One. Gaussian Matrix Ensembles
Chapter Ten. Lattice paths and growth models
Chapter Three. Laguerre And Jacobi Ensembles
Index
Chapter Seven. Scaled limits at β = 1, 2 and 4
Chapter Five. Correlation functions at β = 2
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Chapter Eleven. The Calogero–Sutherland model
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Chapter Two. Circular Ensembles
Chapter Fourteen. Fluctuation formulas and universal behavior of correlations
Contents
Chapter Eight. Eigenvalue probabilities – Painlevé systems approach
Frontmatter --
Chapter Six. Correlation Functions At β= 1 And 4
Chapter Fifteen. The two-dimensional one-component plasma
Preface
Chapter Four. The Selberg Integral
Chapter Nine. Eigenvalue probabilities— Fredholm determinant approach
Bibliography