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E-grāmata: Log-Gases and Random Matrices (LMS-34)

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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; Painleve 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.

Recenzijas

"Log-Gases and Random Matrices is an excellent book. It is bound to become an instant classic and the standard reference to a large body of contemporary random matrix theory. It is a well-written tour through a vast landscape. The contemporary literature is extensively referenced and incorporated in the text, and the material is presented from several perspectives. Forrester has achieved the pedagogical equivalent of Dyson's 'Threefold Way' by writing an advanced monograph appealing equally to physicists, mathematicians, and statisticians."--Steven Joel Miller and Eduardo Duenez, Mathematical Reviews

Papildus informācija

Encyclopedic in scope, this book achieves an excellent balance between the theoretical and physical approaches to the subject. It coherently leads the reader from first-principle definitions, through a combination of physical and mathematical arguments, to the full derivation of many fundamental results. The vast amount of material and impeccable choice of topics make it an invaluable reference. -- Eduardo Duenez, University of Texas, San Antonio This self-contained treatment starts from the basics and leads to the 'high end' of the subject. Forrester often gives new derivations of old results that beginners will find helpful, and the coverage of comprehensive topics will be useful to practitioners in the field. -- Boris Khoruzhenko, Queen Mary, University of London
Preface v
Chapter 1 Gaussian matrix ensembles
1(52)
1.1 Random real symmetric matrices
1(4)
1.2 The eigenvalue p.d.f. for the GOE
5(6)
1.3 Random complex Hermitian and quaternion real Hermitian matrices
11(9)
1.4 Coulomb gas analogy
20(10)
1.5 High-dimensional random energy landscapes
30(3)
1.6 Matrix integrals and combinatorics
33(8)
1.7 Convergence
41(1)
1.8 The shifted mean Gaussian ensembles
42(1)
1.9 Gaussian β-ensemble
43(10)
Chapter 2 Circular ensembles
53(32)
2.1 Scattering matrices and Floquet operators
53(3)
2.2 Definitions and basic properties
56(5)
2.3 The elements of a random unitary matrix
61(5)
2.4 Poisson kernel
66(2)
2.5 Cauchy ensemble
68(3)
2.6 Orthogonal and symplectic unitary random matrices
71(2)
2.7 Log-gas systems with periodic boundary conditions
73(3)
2.8 Circular β-ensemble
76(5)
2.9 Real orthogonal β-ensemble
81(4)
Chapter 3 Laguerre and Jacobi ensembles
85(48)
3.1 Chiral random matrices
85(5)
3.2 Wishart matrices
90(8)
3.3 Further examples of the Laguerre ensemble in quantum mechanics
98(8)
3.4 The eigenvalue density
106(4)
3.5 Correlated Wishart matrices
110(1)
3.6 Jacobi ensemble and Wishart matrices
111(4)
3.7 Jacobi ensemble and symmetric spaces
115(3)
3.8 Jacobi ensemble and quantum conductance
118(7)
3.9 A circular Jacobi ensemble
125(2)
3.10 Laguerre β-ensemble
127(2)
3.11 Jacobi β-ensemble
129(1)
3.12 Circular Jacobi β-ensemble
130(3)
Chapter 4 The Selberg integral
133(53)
4.1 Selberg's derivation
133(4)
4.2 Anderson's derivation
137(8)
4.3 Consequences for the β-ensembles
145(11)
4.4 Generalization of the Dixon-Anderson integral
156(4)
4.5 Dotsenko and Fateev's derivation
160(5)
4.6 Aomoto's derivation
165(7)
4.7 Normalization of the eigenvalue p.d.f.'s
172(8)
4.8 Free energy
180(6)
Chapter 5 Correlation functions at β = 2
186(50)
5.1 Successive integrations
186(7)
5.2 Functional differentiation and integral equation approaches
193(4)
5.3 Ratios of characteristic polynomials
197(3)
5.4 The classical weights
200(7)
5.5 Circular ensembles and the classical groups
207(5)
5.6 Log-gas systems with periodic boundary conditions
212(5)
5.7 Partition function in the case of a general potential
217(6)
5.8 Biorthogonal structures
223(6)
5.9 Determinantal k-component systems
229(7)
Chapter 6 Correlation functions at β = 1 and 4
236(47)
6.1 Correlation functions at β = 4
236(10)
6.2 Construction of the skew orthogonal polynomials at β = 4
246(5)
6.3 Correlation functions at β = 1
251(12)
6.4 Construction of the skew orthogonal polynomials and summation formulas
263(6)
6.5 Alternate correlations at β = 1
269(5)
6.6 Superimposed β = 1 systems
274(4)
6.7 A two-component log-gas with charge ratio 1:2
278(5)
Chapter 7 Scaled limits at β = 1, 2 and 4
283(45)
7.1 Scaled limits at β = 2---Gaussian ensembles
283(7)
7.2 Scaled limits at β = 2---Laguerre and Jacobi ensembles
290(7)
7.3 Log-gas systems with periodic boundary conditions
297(1)
7.4 Asymptotic behavior of the one- and two-point functions at β = 2
298(3)
7.5 Bulk scaling and the zeros of the Riemann zeta function
301(7)
7.6 Scaled limits at β = 4---Gaussian ensemble
308(4)
7.7 Scaled limits at β = 4---Laguerre and Jacobi ensembles
312(4)
7.8 Scaled limits at β = 1---Gaussian ensemble
316(3)
7.9 Scaled limits at β = 1---Laguerre and Jacobi ensembles
319(4)
7.10 Two-component log-gas with charge ratio 1:2
323(5)
Chapter 8 Eigenvalue probabilities---Painleve systems approach
328(52)
8.1 Definitions
328(5)
8.2 Hamiltonian formulation of the Painleve theory
333(16)
8.3 σ-form Painleve equation characterizations
349(14)
8.4 The cases β = 1 and 4---circular ensembles and bulk
363(9)
8.5 Discrete Painleve equations
372(3)
8.6 Orthogonal polynomial approach
375(5)
Chapter 9 Eigenvalue probabilities---Fredholm determinant approach
380(60)
9.1 Fredholm determinants
380(5)
9.2 Numerical computations using Fredholm determinants
385(1)
9.3 The sine kernel
386(7)
9.4 The Airy kernel
393(6)
9.5 Bessel kernels
399(4)
9.6 Eigenvalue expansions for gap probabilities
403(13)
9.7 The probabilities Esoftβ (n; (s, ∞)) forβ = 1, 4
416(5)
9.8 The probabilities Ehardβ (n; (O, s); a) forβ = 1, 4
421(5)
9.9 Riemann-Hilbert viewpoint
426(9)
9.10 Nonlinear equations from the Virasoro constraints
435(5)
Chapter 10 Lattice paths and growth models
440(65)
10.1 Counting formulas for directed nonintersecting paths
440(16)
10.2 Dimers and tilings
456(7)
10.3 Discrete polynuclear growth model
463(8)
10.4 Further interpretations and variants of the RSK correspondence
471(9)
10.5 Symmetrized growth models
480(7)
10.6 The Hammersley process
487(5)
10.7 Symmetrized permutation matrices
492(3)
10.8 Gap probabilities and scaled limits
495(5)
10.9 Hammersley process with sources on the boundary
500(5)
Chapter 11 The Calogero-Sutherland model
505(38)
11.1 Shifted mean parameter-dependent Gaussian random matrices
505(7)
11.2 Other parameter-dependent ensembles
512(4)
11.3 The Calogero-Sutherland quantum systems
516(5)
11.4 The Schrodinger operators with exchange terms
521(3)
11.5 The operators H(H, Ex), H(L, Ex) and H(J, Ex)
524(6)
11.6 Dynamical correlations for β = 2
530(10)
11.7 Scaled limits
540(3)
Chapter 12 Jack polynomials
543(49)
12.1 Nonsymmetric Jack polynomials
543(7)
12.2 Recurrence relations
550(3)
12.3 Application of the recurrences
553(2)
12.4 A generalized binomial theorem and an integration formula
555(3)
12.5 Interpolation nonsymmetric Jack polynomials
558(6)
12.6 The symmetric Jack polynomials
564(15)
12.7 Interpolation symmetric Jack polynomials
579(4)
12.8 Pieri formulas
583(9)
Chapter 13 Correlations for general β
592(66)
13.1 Hypergeometric functions and Selberg correlation integrals
592(9)
13.2 Correlations at even β
601(12)
13.3 Generalized classical polynomials
613(14)
13.4 Green functions and zonal polynomials
627(6)
13.5 Inter-relations for spacing distributions
633(1)
13.6 Stochastic differential equations
634(6)
13.7 Dynamical correlations in the circular β ensemble
640(18)
Chapter 14 Fluctuation formulas and universal behavior of correlations
658(43)
14.1 Perfect screening
658(5)
14.2 Macroscopic balance and density
663(2)
14.3 Variance of a linear statistic
665(7)
14.4 Gaussian fluctuations of a linear statistic
672(8)
14.5 Charge and potential fluctuations
680(8)
14.6 Asymptotic properties of Eβ(n; J) and Pβ(n; J)
688(10)
14.7 Dynamical correlations
698(3)
Chapter 15 The two-dimensional one-component plasma
701(64)
15.1 Complex random matrices and polynomials
701(5)
15.2 Quantum particles in a magnetic field
706(5)
15.3 Correlation functions
711(7)
15.4 General properties of the correlations and fluctuation formulas
718(7)
15.5 Spacing distributions
725(4)
15.6 The sphere
729(9)
15.7 The pseudosphere
738(6)
15.8 Metallic boundary conditions
744(3)
15.9 Antimetallic boundary conditions
747(5)
15.10 Eigenvalues of real random matrices
752(8)
15.11 Classification of non-Hermitian random matrices
760(5)
Bibliography 765(20)
Index 785
Peter J. Forrester is professor of mathematics at the University of Melbourne.
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