Discrete orthogonal polynomials: asymptotics and applications
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Princeton [u.a.]
Princeton University Press
2007
|
| Series: | Annals of mathematics studies
164 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VI, 170 S. Ill., graph. Darst. |
| ISBN: | 0691127336 9780691127330 0691127344 9780691127347 |
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| 245 | 1 | 0 | |a Discrete orthogonal polynomials |b asymptotics and applications |c J. Baik ... [et al.] |
| 264 | 1 | |a Princeton [u.a.] |b Princeton University Press |c 2007 | |
| 300 | |a VI, 170 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Annals of mathematics studies |v 164 | |
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Record in the Search Index
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| adam_text | Contents
Preface vij
Chapter 1. Introduction 1
1.1 Motivating applications 1
1.2 Discrete orthogonal polynomials 8
1.3 Assumptions 10
1.4 Goals and methodology n
1.5 Outline of the rest of the book 22
1.6 Research background 23
Chapter 2. Asymptotics of General Discrete Orthogonal Polynomials in the Complex Plane 25
2.1 The equilibrium energy problem 25
2.2 Elements of hyperelliptic function theory 31
2.3 Results on asymptotics of discrete orthogonal polynomials 33
2.4 Equilibrium measures for some classical discrete orthogonal polynomials 41
Chapter 3. Applications 49
3.1 Discrete orthogonal polynomial ensembles and their particle statistics 49
3.2 Dual ensembles and hole statistics 51
3.3 Results on asymptotic universality for general weights 52
3.4 Random rhombus tilings of a hexagon 57
3.5 The continuum limit of the Toda lattice 60
Chapter 4. An Equivalent Riemann Hilbert Problem 67
4.1 Choice of A: the transformation from P(z; N, k) to Q(z: N. k) 67
4.2 Removal of poles in favor of discontinuities along contours: the transformation
from Q(z; N, k) to R(z) 69
4.3 Use of the equilibrium measure: the transformation from R(z) to S(z) 70
4.4 Steepest descent: the transformation from S(z) to X(z) 78
4.5 Properties of X(z) 79
Chapter 5. Asymptotic Analysis 87
5.1 Construction of a global parametrix for X(z) 87
5.2 Error estimation 99
Chapter 6. Discrete Orthogonal Polynomials: Proofs of Theorems Stated in §2.3 105
6.1 Asymptotic analysis of P(z; N, k) for z 6 C [a. b] 105
6.2 Asymptotic behavior of irN.k (z) for 2 near a void of [a. b : the proof of Theorem 2.9 107
6.3 Asymptotic behavior of iTN.k(z) for z near a saturated region of [a. 6] 108
6.4 Asymptotic behavior of irN.k(z) for z near a band 110
6.5 Asymptotic behavior of 7r,v.t(z) for z near a band edge 112
Chapter 7. Universality: Proofs of Theorems Stated in §3.3 115
7.1 Relation between correlation functions of dual ensembles 115
vi CONTENTS
7.2 Exact formulae for KN,k(x, y) 118
7.3 Asymptotic formulae for ifjv,k {x, y) and universality 124
Appendix A. The Explicit Solution of Riemann Hilbert Problem 5.1 135
A.I Steps for making the jump matrix piecewise constant: the transformation from X(z) to Y (z) 135
A.2 Construction of Y*(z) using hyperelliptic function theory 137
A.3 The matrix X(z) and its properties 141
Appendix B. Construction of the Hahn Equilibrium Measure: the Proof of Theorem 2.17 145
B.I General strategy: the one band ansatz 145
B.2 The void band void configuration 146
B.3 The saturated band void configuration 149
B.4 The void band saturated configuration 150
B.5 The saturated band saturated configuration 151
Appendix C. List of Important Symbols 153
Bibliography 163
Index 167
|
| adam_txt |
Contents
Preface vij
Chapter 1. Introduction 1
1.1 Motivating applications 1
1.2 Discrete orthogonal polynomials 8
1.3 Assumptions 10
1.4 Goals and methodology n
1.5 Outline of the rest of the book 22
1.6 Research background 23
Chapter 2. Asymptotics of General Discrete Orthogonal Polynomials in the Complex Plane 25
2.1 The equilibrium energy problem 25
2.2 Elements of hyperelliptic function theory 31
2.3 Results on asymptotics of discrete orthogonal polynomials 33
2.4 Equilibrium measures for some classical discrete orthogonal polynomials 41
Chapter 3. Applications 49
3.1 Discrete orthogonal polynomial ensembles and their particle statistics 49
3.2 Dual ensembles and hole statistics 51
3.3 Results on asymptotic universality for general weights 52
3.4 Random rhombus tilings of a hexagon 57
3.5 The continuum limit of the Toda lattice 60
Chapter 4. An Equivalent Riemann Hilbert Problem 67
4.1 Choice of A: the transformation from P(z; N, k) to Q(z: N. k) 67
4.2 Removal of poles in favor of discontinuities along contours: the transformation
from Q(z; N, k) to R(z) 69
4.3 Use of the equilibrium measure: the transformation from R(z) to S(z) 70
4.4 Steepest descent: the transformation from S(z) to X(z) 78
4.5 Properties of X(z) 79
Chapter 5. Asymptotic Analysis 87
5.1 Construction of a global parametrix for X(z) 87
5.2 Error estimation 99
Chapter 6. Discrete Orthogonal Polynomials: Proofs of Theorems Stated in §2.3 105
6.1 Asymptotic analysis of P(z; N, k) for z 6 C \ [a. b] 105
6.2 Asymptotic behavior of irN.k (z) for 2 near a void of [a. b\: the proof of Theorem 2.9 107
6.3 Asymptotic behavior of iTN.k(z) for z near a saturated region of [a. 6] 108
6.4 Asymptotic behavior of irN.k(z) for z near a band 110
6.5 Asymptotic behavior of 7r,v.t(z) for z near a band edge 112
Chapter 7. Universality: Proofs of Theorems Stated in §3.3 115
7.1 Relation between correlation functions of dual ensembles 115
vi CONTENTS
7.2 Exact formulae for KN,k(x, y) 118
7.3 Asymptotic formulae for ifjv,k {x, y) and universality 124
Appendix A. The Explicit Solution of Riemann Hilbert Problem 5.1 135
A.I Steps for making the jump matrix piecewise constant: the transformation from X(z) to Y"(z) 135
A.2 Construction of Y*(z) using hyperelliptic function theory 137
A.3 The matrix X(z) and its properties 141
Appendix B. Construction of the Hahn Equilibrium Measure: the Proof of Theorem 2.17 145
B.I General strategy: the one band ansatz 145
B.2 The void band void configuration 146
B.3 The saturated band void configuration 149
B.4 The void band saturated configuration 150
B.5 The saturated band saturated configuration 151
Appendix C. List of Important Symbols 153
Bibliography 163
Index 167 |
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| isbn | 0691127336 9780691127330 0691127344 9780691127347 |
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| physical | VI, 170 S. Ill., graph. Darst. |
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| spelling | Discrete orthogonal polynomials asymptotics and applications J. Baik ... [et al.] Princeton [u.a.] Princeton University Press 2007 VI, 170 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies 164 Polynômes orthogonaux - Théorie asymptotique Orthogonal polynomials Asymptotic theory Orthogonale Polynome (DE-588)4172863-4 gnd rswk-swf Asymptotik (DE-588)4126634-1 gnd rswk-swf Orthogonale Polynome (DE-588)4172863-4 s Asymptotik (DE-588)4126634-1 s DE-604 Baik, J. Sonstige oth Annals of mathematics studies 164 (DE-604)BV000000991 164 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015479332&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Discrete orthogonal polynomials asymptotics and applications Annals of mathematics studies Polynômes orthogonaux - Théorie asymptotique Orthogonal polynomials Asymptotic theory Orthogonale Polynome (DE-588)4172863-4 gnd Asymptotik (DE-588)4126634-1 gnd |
| subject_GND | (DE-588)4172863-4 (DE-588)4126634-1 |
| title | Discrete orthogonal polynomials asymptotics and applications |
| title_auth | Discrete orthogonal polynomials asymptotics and applications |
| title_exact_search | Discrete orthogonal polynomials asymptotics and applications |
| title_exact_search_txtP | Discrete orthogonal polynomials asymptotics and applications |
| title_full | Discrete orthogonal polynomials asymptotics and applications J. Baik ... [et al.] |
| title_fullStr | Discrete orthogonal polynomials asymptotics and applications J. Baik ... [et al.] |
| title_full_unstemmed | Discrete orthogonal polynomials asymptotics and applications J. Baik ... [et al.] |
| title_short | Discrete orthogonal polynomials |
| title_sort | discrete orthogonal polynomials asymptotics and applications |
| title_sub | asymptotics and applications |
| topic | Polynômes orthogonaux - Théorie asymptotique Orthogonal polynomials Asymptotic theory Orthogonale Polynome (DE-588)4172863-4 gnd Asymptotik (DE-588)4126634-1 gnd |
| topic_facet | Polynômes orthogonaux - Théorie asymptotique Orthogonal polynomials Asymptotic theory Orthogonale Polynome Asymptotik |
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