Random matrix theory: invariant ensembles and universality
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Courant Inst. of Math. Sciences [u.a.]
2009
|
| Series: | Courant lecture notes in mathematics
18 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | IX, 217 S. |
| ISBN: | 9780821847374 |
Staff View
MARC
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| 490 | 1 | |a Courant lecture notes in mathematics |v 18 | |
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Record in the Search Index
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| adam_text | Contents
Preface
ix
Part
1.
Invariant Random Matrix Ensembles: Unified Derivation of
Eigenvalue Cluster and Correlation Functions
1
Chapter
1.
Introduction and Examples
3
1.1.
Introduction
3
1.2.
Three Examples
3
1.3.
Synopsis of the Book
5
1.4.
Some General Remarks
6
Chapter
2.
Three Classes of Invariant Ensembles
9
2.1.
Precise Definitions of the Ensembles
9
2.2.
Explicit Form of the Invariant Probability Densities
10
2.3.
Separating Out the Eigenvalue Densities: Matrices with Simple
Eigenvalues Form Open Dense Sets of Full Measure
14
2.4.
Separating Out the Eigenvalue Densities: Computing the Jacobians
18
2.5.
Integrating Out Variables Other Than the Eigenvalues
32
Chapter
3.
Auxiliary Facts from Functional Analysis, Pfaffians,
and Three Integral Identities
37
3.1.
Statement of Auxiliary Results
37
3.2.
Proof of the Results from Functional Analysis
38
3.3.
Proof of the Result on the Pfaffian
42
3.4.
Proof of the Three Integral Identities
57
Chapter
4.
Eigenvalue Statistics for the Three Types of Ensembles
65
4.1.
Computing Expectations: Correlation Kernels
65
4.2.
Computing Gap Probabilities
86
4.3.
Computing Occupational Probabilities
86
4.4.
Computing Correlation and Cluster Functions
89
Part
2.
Universality for Orthogonal and Symplectic Ensembles
113
Chapter
5.
Widom s Formulae for the
β
= 1
and
4
Correlation Kernels
115
5.1.
Statement of Widom s Formulae
115
5.2.
Proof of Widom s Formulae
119
5.3.
Additional Properties of the
β
= 1,4
Correction Terms
130
viii CONTENTS
5.4.
General
Remarks Concerning the Matrices D^ and
е#
and Another Useful Identity among Certain Determinants
135
Chapter
6.
Large
N
Eigenvalue Statistics for the
β
= 1,4
Ensembles
with Monomial Potentials: Universality
139
6.1.
Introduction and Statement of the Results
139
6.2.
Auxiliary Results
146
6.3.
Proofs of Theorem
6.7
and Corollary
6.12 154
6.4.
Asymptotics of OPs: Matching Formulae
163
6.5.
Asymptotics of OPs: Single Integrals and Proof of Theorem
6.55 174
6.6.
Asymptotics of OPs: Double Integrals and Proof of Theorem
6.38 178
6.7.
Convergence of Derivatives and Integrals of the Christoffel-Darboux
Kernel for Monomial Potentials
190
6.8.
Differential Equation for the Density h(x) of the Equilibrium
Measure and Proof of Theorem
6.51 200
Bibliography
211
Index
217
Random Matrix Theory:
Invariant Ensembles and Universality
PERCY DEIFT AND DIMITRI GIOEV
This book features a unified derivation of the mathematical theory of the
three classical types of invariant random matrix ensembles
—
orthogonal,
unitary, and symplectic. The authors follow the approach of Tracy and
Widom,
but the exposition here contains a substantial amount of additional
material, in particular, facts from functional analysis and the theory of
Pfaffians. The main result in the book is a proof of universality for orthog¬
onal and symplectic ensembles corresponding to generalized Gaussian
type weights following the authors prior work. New, quantitative error
estimates are derived.
The book is based in part on a graduate course given by the first author
at the
Courant
Institute in fall
2005.
Subsequently, the second author gave
a modified version of this course at the University of Rochester in spring
2007.
Anyone with some background in complex analysis, probability
theory, and linear algebra and an interest in the mathematical foundations
of random matrix theory will benefit from studying this valuable reference.
For additional information
I and updates on this book, visit
www.ams.org/bookpages/cln-1
8
New York University
ISBN 978-0-8218-A737-4
780821
847374
CLN/18
AMS
on the Web
www.ams.org
|
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| language | English |
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| spelling | Deift, Percy 1945- Verfasser (DE-588)140992995 aut Random matrix theory invariant ensembles and universality Percy Deift ; Dimitri Gioev New York, NY Courant Inst. of Math. Sciences [u.a.] 2009 IX, 217 S. txt rdacontent n rdamedia nc rdacarrier Courant lecture notes in mathematics 18 Includes bibliographical references and index Random matrices Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Stochastische Matrix (DE-588)4057624-3 s DE-604 Gioev, Dimitri 1973- Verfasser (DE-588)139223592 aut Erscheint auch als Online-Ausgabe 978-1-4704-1761-1 Courant lecture notes in mathematics 18 (DE-604)BV012714106 18 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018012829&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018012829&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Deift, Percy 1945- Gioev, Dimitri 1973- Random matrix theory invariant ensembles and universality Courant lecture notes in mathematics Random matrices Stochastische Matrix (DE-588)4057624-3 gnd |
| subject_GND | (DE-588)4057624-3 |
| title | Random matrix theory invariant ensembles and universality |
| title_auth | Random matrix theory invariant ensembles and universality |
| title_exact_search | Random matrix theory invariant ensembles and universality |
| title_full | Random matrix theory invariant ensembles and universality Percy Deift ; Dimitri Gioev |
| title_fullStr | Random matrix theory invariant ensembles and universality Percy Deift ; Dimitri Gioev |
| title_full_unstemmed | Random matrix theory invariant ensembles and universality Percy Deift ; Dimitri Gioev |
| title_short | Random matrix theory |
| title_sort | random matrix theory invariant ensembles and universality |
| title_sub | invariant ensembles and universality |
| topic | Random matrices Stochastische Matrix (DE-588)4057624-3 gnd |
| topic_facet | Random matrices Stochastische Matrix |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018012829&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018012829&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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