Log-gases and random matrices:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton, NJ [u.a.]
Princeton Univ. Press
2010
|
| Series: | The London Mathematical Society monographs series
34 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 791 S. graph. Darst. |
| ISBN: | 9780691128290 0691128294 |
Staff View
MARC
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| 100 | 1 | |a Forrester, Peter J. |e Verfasser |0 (DE-588)142224413 |4 aut | |
| 245 | 1 | 0 | |a Log-gases and random matrices |c P. J. Forrester |
| 264 | 1 | |a Princeton, NJ [u.a.] |b Princeton Univ. Press |c 2010 | |
| 300 | |a XIV, 791 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Record in the Search Index
| _version_ | 1804143672587976704 |
|---|---|
| adam_text | Contents
Preface
v
Chapter
1.
Gaussian matrix ensembles
1
1.1
Random real symmetric matrices
1
1.2
The eigenvalue p.d.f. for the GOE
5
1.3
Random complex Hermitian and quaternion real Hermitian matrices
11
1.4
Coulomb gas analogy
20
1.5
High-dimensional random energy landscapes
30
1.6
Matrix integrals and combinatorics
33
1.7
Convergence
41
1.8
The shifted mean Gaussian ensembles
42
1.9
Gaussian /J-ensemble
43
Chapter
2.
Circular ensembles
53
2.1
Scattering matrices and Floquet operators
53
2.2
Definitions and basic properties
56
2.3
The elements of a random unitary matrix
61
2.4
Poisson
kernel
66
2.5
Cauchy ensemble
68
2.6
Orthogonal and symplectic unitary random matrices
71
2.7
Log-gas systems with periodic boundary conditions
73
2.8
Circular /3-ensemble
76
2.9
Real orthogonal /3-ensemble
81
Chapter
3.
Laguerre and Jacobi ensembles
85
3.1
Chiral random matrices
85
3.2
Wishart
matrices
90
3.3
Further examples of the Laguerre ensemble in quantum mechanics
98
3.4
The eigenvalue density
106
3.5
Correlated
Wishart
matrices
110
3.6
Jacobi ensemble and
Wishart
matrices
111
3.7
Jacobi ensemble and symmetric spaces
115
3.8
Jacobi ensemble and quantum conductance
118
3.9
A circular Jacobi ensemble
125
3.10
Laguerre/3-ensemble
127
3.11
Jacobin-ensemble
129
3.12
Circular Jacobi/S-ensemble
130
Chapter
4.
The Selberg integral
133
4.1
Selberg s derivation
133
4.2
Anderson s derivation
137
4.3
Consequences for the
/ö-ensembles
145
XII CONTENTS
4.4
Generalization of the Dixon-
Anderson
integral
156
4.5
Dotsenko and Fateev s derivation
160
4.6
Aomoto s derivation
165
4.7
Normalization of the eigenvalue p.d.f. s
172
4.8
Free energy
180
Chapter
5.
Correlation functions at
0 = 2 186
5.1
Successive integrations
186
5.2
Functional differentiation and integral equation approaches
193
5.3
Ratios of characteristic polynomials
197
5.4
The classical weights
200
5.5
Circular ensembles and the classical groups
207
5.6
Log-gas systems with periodic boundary conditions
212
5.7
Partition function in the case of a general potential
217
5.8 Biorthogonal
structures
223
5.9
Determinantal fc-component systems
229
Chapter
6.
Correlation functions at
β
= 1
and
4 236
6.1
Correlation functions at
β
= 4 236
6.2
Construction of the skew orthogonal polynomials at
β
= 4 246
6.3
Correlation functions at
β
= 1 251
6.4
Construction of the skew orthogonal polynomials and summation formulas
263
6.5
Alternate correlations at
β
— 1 269
6.6
Superimposed
β
= 1
systems
274
6.7
A two-component log-gas with charge ratio
1:2 278
Chapter
7.
Scaled limits at
β
= 1,2
and
4 283
7.1
Scaled limits at
β
= 2 —
Gaussian ensembles
283
7.2
Scaled limits at
β
= 2 —
Laguerre and Jacobi ensembles
290
7.3
Log-gas systems with periodic boundary conditions
297
7.4
Asymptotic behavior of the one- and two-point functions at
β
= 2 298
7.5
Bulk scaling and the zeros of the Riemann
zeta
function
301
7.6
Scaled limits at
β
= 4 —
Gaussian ensemble
308
7.7
Scaled limits at
β
= 4 —
Laguerre and Jacobi ensembles
312
7.8
Scaled limits at
β
= 1 —
Gaussian ensemble
316
7.9
Scaled limits at
β
= 1 —
Laguerre and Jacobi ensembles
319
7.10
Two-component log-gas with charge ratio
1
:2
323
Chapter
8.
Eigenvalue probabilities
—
Painlevé
systems approach
328
8.1
Definitions
328
8.2
Hamiltonian formulation of the
Painlevé
theory
333
8.3
σ
-form
Painlevé
equation characterizations
349
8.4
The cases
/3 = 1
and
4 —
circular ensembles and bulk
З6З
8.5
Discrete
Painlevé
equations
372
8.6
Orthogonal polynomial approach
375
ChapterQ. Eigenvalue probabilities
—
Fredholm
determinant approach
380
9.1
Fredholm
determinants
330
9.2
Numerical computations using
Fredholm
determinants
385
9.3
The sine kernel
386
9.4
The Airy kernel
393
9.5
Bessel kernels
399
CONTENTS
xjj¡
9.6
Eigenvalue expansions for gap probabilities
4О3
9.7
The probabilities Ebeoh(n; (s,
00))
for
β
= 1,4 416
9.8
The probabilities £$Brd(n;
(0,
s); a) for
β
= 1,4 421
9.9
Riemann-Hilbert viewpoint
426
9.10
Nonlinear equations from the Virasoro constraints
435
Chapter
10.
Lattice paths and growth models
44O
10.1
Counting formulas for directed nonintersecting paths
440
10.2
Dimers and tilings
456
10.3
Discrete polynuclear growth model
463
10.4
Further interpretations and variants of the RSK correspondence
471
10.5
Symmetrized growth models
480
10.6
The Hammersley process
437
10.7
Symmetrized permutation matrices
492
10.8
Gap probabilities and scaled limits
495
10.9
Hammersley process with sources on the boundary
500
Chapter
11.
The Calogero-Sutherland model 505
11.1
Shifted mean parameter-dependent Gaussian random matrices
505
11.2
Other parameter-dependent ensembles
512
11.3
The Calogero-Sutherland quantum systems 5I6
11.4
The
Schrödinger
operators with exchange terms
521
11.5
The operators
//(HEx),//(L Ex)
and
Я ·7^)
524
11.6
Dynamical correlations for
β
= 2 530
11.7
Scaled limits
54g
Chapter^. Jack polynomials
543
12.1
Nonsymmetric Jack polynomials
543
12.2
Recurrence relations
55O
12.3
Application of the recurrences
553
12.4
A generalized binomial theorem and an integration formula
555
12.5
Interpolation nonsymmetric Jack polynomials
558
12.6
The symmetric Jack polynomials
554
12.7
Interpolation symmetric Jack polynomials
579
12.8
Pieri
formulas
583
Chapter
13.
Correlations for general
β
592
13.1
Hypergeometric functions and Selberg correlation integrals
592
13.2
Correlations at even
β
601
13.3
Generalized classical polynomials
613
13.4
Green functions and zonal polynomials
627
13.5
Inter-relations for spacing distributions
633
13.6
Stochastic differential equations
634
13.7
Dynamical correlations in the circular
β
ensemble
640
Chapter
14.
Fluctuation formulas and universal behavior of correlations
658
14.1
Perfect screening
658
14.2
Macroscopic balance and density
663
14.3
Variance of a linear statistic
665
14.4
Gaussian fluctuations of a linear statistic
672
14.5
Charge and potential fluctuations
680
14.6
Asymptotic properties of Eg(n;J) and Pg(n; J)
688
XIV
CONTENTS
14.7
Dynamical correlations
698
Chapter
15.
The two-dimensional one-component plasma
701
15.1
Complex random matrices and polynomials
701
15.2
Quantum particles in a magnetic field
706
15.3
Correlation functions
711
15.4
General properties of the correlations and fluctuation formulas
718
15.5
Spacing distributions
725
15.6
The sphere
729
15.7
The pseudosphere
738
15.8
Metallic boundary conditions
744
15.9
Antimetallic boundary conditions
747
15.10
Eigenvalues of real random matrices
752
15.11
Classification of non-Hermitian random matrices 7gg
Bibliography
755
Index
785
|
| any_adam_object | 1 |
| author | Forrester, Peter J. |
| author_GND | (DE-588)142224413 |
| author_facet | Forrester, Peter J. |
| author_role | aut |
| author_sort | Forrester, Peter J. |
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| building | Verbundindex |
| bvnumber | BV036951858 |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)664682101 (DE-599)BSZ327191015 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036951858 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:51:23Z |
| institution | BVB |
| isbn | 9780691128290 0691128294 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020866902 |
| oclc_num | 664682101 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-188 DE-703 |
| owner_facet | DE-19 DE-BY-UBM DE-188 DE-703 |
| physical | XIV, 791 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Princeton Univ. Press |
| record_format | marc |
| series | The London Mathematical Society monographs series |
| series2 | The London Mathematical Society monographs series |
| spelling | Forrester, Peter J. Verfasser (DE-588)142224413 aut Log-gases and random matrices P. J. Forrester Princeton, NJ [u.a.] Princeton Univ. Press 2010 XIV, 791 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier The London Mathematical Society monographs series 34 Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Statistische Physik (DE-588)4057000-9 gnd rswk-swf Random matrices Stochastische Matrix (DE-588)4057624-3 s Statistische Physik (DE-588)4057000-9 s DE-604 The London Mathematical Society monographs series 34 (DE-604)BV045355493 34 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020866902&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Forrester, Peter J. Log-gases and random matrices The London Mathematical Society monographs series Stochastische Matrix (DE-588)4057624-3 gnd Statistische Physik (DE-588)4057000-9 gnd |
| subject_GND | (DE-588)4057624-3 (DE-588)4057000-9 |
| title | Log-gases and random matrices |
| title_auth | Log-gases and random matrices |
| title_exact_search | Log-gases and random matrices |
| title_full | Log-gases and random matrices P. J. Forrester |
| title_fullStr | Log-gases and random matrices P. J. Forrester |
| title_full_unstemmed | Log-gases and random matrices P. J. Forrester |
| title_short | Log-gases and random matrices |
| title_sort | log gases and random matrices |
| topic | Stochastische Matrix (DE-588)4057624-3 gnd Statistische Physik (DE-588)4057000-9 gnd |
| topic_facet | Stochastische Matrix Statistische Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020866902&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV045355493 |
| work_keys_str_mv | AT forresterpeterj loggasesandrandommatrices |