Spin glasses: a challenge for mathematicians: cavity and mean field models
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2003
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete
Folge 3 ; 46 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 586 S. |
| ISBN: | 3540003568 |
Staff View
MARC
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| 245 | 1 | 0 | |a Spin glasses: a challenge for mathematicians |b cavity and mean field models |c Michel Talgrand |
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Record in the Search Index
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| adam_text | Contents
0. Introduction 1
1. A Toy Model, the REM 7
1.1 High Temperature 7
1.2 Low Temperature 13
1.3 Notes and Comments 26
2. The Sherrington-Kirkpatrick Model 27
2.1 Introduction 27
2.2 The SK Model with no External Field at High Temperature . 30
2.3 The Cavity Method 49
2.4 The Replica-Symmetric Solution and the Smart Path Method 72
2.5 Exponential Inequalities 98
2.6 Second Moment Computations
and the Almeida-Thouless Line 103
2.7 Central Limit Theorems 114
2.8 Replica-Symmetry Breaking 124
2.9 Guerra s Replica Symmetric Bound 130
2.10 Guerra-Toninelli Interpolation
and the Thermodynarnical Limit 140
2.11 Guerra s Broken Replica-Symmetry Bound
and the Parisi Formula 143
2.12 The Ghirlanda-Guerra Identities 157
2.13 The SK Model with 2-Components Spins 162
2.14 A Research Problem: The Transition at 0 = 1 182
2.15 The Physicist s Replica Method 191
2.16 Notes and Comments 195
3. The Capacity of the Perceptron: The Ising Case 201
3.1 Introduction 201
3.2 The Replica-Symmetric Equations and the Smart Path 203
3.3 The Main Estimate 208
3.4 The Replica-Symmetric Solution 219
VIII Contents
3.5 Exponential Inequalities 226
3.6 Central Limit Theorems 234
3.7 An Approximation Procedure 247
3.8 The Bernoulli Perceptron and Approximate Integration
by Parts 253
3.9 Estimates for the Unbounded Case 258
3.10 The Gardner Formula for Small a 268
3.11 The Gardner Formula in the Bernoulli Case 280
3.12 Notes and Comments 287
4. Capacity of the Perceptron: The Gaussian
and the Spherical Case 289
4.1 The Shcherbina and Tirozzi Model 289
4.2 The Replica-Symmetric Equations 296
4.3 Approximations of the Limiting Case 302
4.4 The Gardner Formula for the Sphere 313
4.5 Notes and Comments 319
5. The Hopfield Model 323
5.1 Introduction: The Curie-Weiss Model 323
5.2 The Replica-Symmetric Equations 329
5.3 The Bovier-Gayrard Localization Theorem 338
5.4 Localization on Balls with Random Centers 358
5.5 Selecting a State with an External Field 371
5.6 Controlling mk((r), k 2 376
5.7 The Smart Path 380
5.8 Integration by Parts 390
5.9 The Replica-Symmetric Solution 394
5.10 Higher Moments and Central Limit Theorems 407
5.11 The p-Spin Hopfield Model 414
5.12 Notes and Comments 429
6. The p-Spin Interaction Model at Low Temperature 431
6.1 A priori Estimates 431
6.2 Construction and Separation of the Lumps 440
6.3 Pure States 448
6.4 Perturbed Hamiltonian
and Extended Ghirlanda-Guerra Identities 453
6.5 One Step of Replica-Symmetry Breaking 466
6.6 A Positivity Principle 485
6.7 Computing the Free Energy 490
6.8 Notes and Comments 497
Contents IX
7. The Diluted SK Model and the K-Sat Problem 499
7.1 Introduction 499
7.2 Pure State 501
7.3 The Functional Order Parameter 511
7.4 The Replica-Symmetric Solution 517
8. An Assignment Problem 537
8.1 Introduction 537
8.2 The Cavity Method 538
8.3 Decoupling 541
8.4 Empirical Measures 548
8.5 Structure Theorems 553
A. Appendix. Elements of Probability Theory 563
A.I Tail Estimates 563
A.2 Bernstein s Inequality 566
A.3 e-Nets 568
A.4 Random Matrices 569
A.5 Poisson Point Processes 572
A.6 Gaussian r.v 573
A.7 Integration by Parts 575
References 577
Index 585
|
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| indexdate | 2024-07-09T21:37:43Z |
| institution | BVB |
| isbn | 3540003568 |
| language | English |
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| physical | IX, 586 S. |
| publishDate | 2003 |
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| publisher | Springer |
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| series | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 |
| spelling | Talagrand, Michel 1952- Verfasser (DE-588)112924379 aut Spin glasses: a challenge for mathematicians cavity and mean field models Michel Talgrand Berlin [u.a.] Springer 2003 IX, 586 S. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : Folge 3 46 Spinglas (DE-588)4138228-6 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Spinglas (DE-588)4138228-6 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Ergebnisse der Mathematik und ihrer Grenzgebiete Folge 3 ; 46 (DE-604)BV000899194 46 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017478440&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Talagrand, Michel 1952- Spin glasses: a challenge for mathematicians cavity and mean field models Ergebnisse der Mathematik und ihrer Grenzgebiete Spinglas (DE-588)4138228-6 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4138228-6 (DE-588)4114528-8 |
| title | Spin glasses: a challenge for mathematicians cavity and mean field models |
| title_auth | Spin glasses: a challenge for mathematicians cavity and mean field models |
| title_exact_search | Spin glasses: a challenge for mathematicians cavity and mean field models |
| title_full | Spin glasses: a challenge for mathematicians cavity and mean field models Michel Talgrand |
| title_fullStr | Spin glasses: a challenge for mathematicians cavity and mean field models Michel Talgrand |
| title_full_unstemmed | Spin glasses: a challenge for mathematicians cavity and mean field models Michel Talgrand |
| title_short | Spin glasses: a challenge for mathematicians |
| title_sort | spin glasses a challenge for mathematicians cavity and mean field models |
| title_sub | cavity and mean field models |
| topic | Spinglas (DE-588)4138228-6 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Spinglas Mathematisches Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017478440&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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