Mean Field Models for Spin Glasses (eBook)

Volume I: Basic Examples

Michel Talagrand (Autor)

eBook Download: PDF

2010 | 2nd rev. and enlarged ed. 2011
XVIII, 485 Seiten
Springer Berlin (Verlag)
978-3-642-15202-3 (ISBN)

Infos

This is a new, completely revised, updated and enlarged edition of the author's Ergebnisse vol. 46: 'Spin Glasses: A Challenge for Mathematicians'. This new edition will appear in two volumes, the present first volume presents the basic results and methods, the second volume is expected to appear in 2011. In the eighties, a group of theoretical physicists introduced several models for certain disordered systems, called 'spin glasses'. These models are simple and rather canonical random structures, of considerable interest for several branches of science (statistical physics, neural networks and computer science). The physicists studied them by non-rigorous methods and predicted spectacular behaviors. This book introduces in a rigorous manner this exciting new area to the mathematically minded reader. It requires no knowledge whatsoever of any physics. The first volume of this new and completely rewritten edition presents six fundamental models and the basic techniques to study them.

M. Talagrand is Research Director in CNRS, and Member of the French Academy of Sciences since 2004.

Inhalt

Contents 6
Introduction 9
The Sherrington-Kirkpatrick Model 17
Introduction 17
Notations and Simple Facts 21
Gaussian Interpolation and the Smart Path Method 28
Latala's Argument 45
A Kind of Central Limit Theorem 67
The Cavity Method 69
Gibbs' Measure the TAP Equations
Second Moment Computations and the Almeida-Thouless Line 96
Beyond the AT Line 109
Central Limit Theorem for the Overlaps 114
Non Gaussian Behavior: Hanen's Theorem 129
The SK Model with d-component Spins 141
The Physicist's Replica Method 156
Notes and Comments 162
The Perceptron Model 166
Introduction 166
The Smart Path 169
Cavity in M 175
The Replica Symmetric Solution 188
Exponential Inequalities 197
Notes and Comments 205
The Shcherbina and Tirozzi Model 206
The Power of Convexity 206
The Replica-Symmetric Equations 222
Controlling the Solutions of the RS Equations 239
Notes and Comments 250
The Hopfield Model 252
Introduction: The Curie-Weiss Model 252
Local Convexity and the Hubbard-Stratonovitch Transform 259
The Bovier-Gayrard Localization Theorem 269
Selecting a State with an External Field 287
Controlling the Overlaps 289
Approximate Integration by Parts and the Replica-Symmetric Equations 299
Notes and Comments 310
The V-statistics Model 312
Introduction 312
The Smart Path 314
Cavity in M 317
The New Equation 321
The Replica-Symmetric Solution 327
The Diluted SK Model and the K-Sat Problem 339
Introduction 339
Pure State 342
The Functional Order Parameter 354
The Replica-Symmetric Solution 359
The Franz-Leone Bound 373
Continuous Spins 378
The Power of Convexity 390
Notes and Comments 409
An Assignment Problem 410
Introduction 410
Overview of the Proof 414
The Cavity Method 415
Decoupling 418
Empirical Measures 426
Operators 435
Notes and Comments 446
Appendix: Elements of Probability Theory 447
How to Use this Appendix 447
Differentiation Inside an Expectation 447
Gaussian Random Variables 449
Gaussian Integration by Parts 452
Tail Estimates 453
How to Use Tail Estimates 456
Bernstein's Inequality 458
-Nets 460
Random Matrices 461
Poisson Random Variables and Point Processes 466
Distances Between Probability Measures 468
The Paley-Zygmund Inequality 472
Differential Inequalities 472
The Latala-Guerra Lemma 473
Proof of Theorem 3.1.4 474
References 476
Index 485
Glossary 488

Produkt-Details

• Erscheint lt. Verlag: 10.11.2010
• Reihe/Serie: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
• Reihe/Serie: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
• Zusatzinfo: XVIII, 485 p.
• Verlagsort: Berlin
• Sprache: englisch
• Original-Titel: Spin Glasses: A Challenge for Mathematicians
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Disordered Systems • Hopfield Model • Mean Field Models • Random Structures • Sherrington Kirkpatrick Model
• ISBN-10: 3-642-15202-3 / 3642152023
• ISBN-13: 978-3-642-15202-3 / 9783642152023