Information, Physics, and Computation available in Hardcover
Information, Physics, and Computation
- ISBN-10:
- 019857083X
- ISBN-13:
- 9780198570837
- Pub. Date:
- 03/27/2009
- Publisher:
- Oxford University Press, USA
- ISBN-10:
- 019857083X
- ISBN-13:
- 9780198570837
- Pub. Date:
- 03/27/2009
- Publisher:
- Oxford University Press, USA
Information, Physics, and Computation
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Overview
This book presents a unified approach to a rich and rapidly evolving research domain at the interface between statistical physics, theoretical computer science/discrete mathematics, and coding/information theory. It is accessible to graduate students and researchers without a specific training in any of these fields. The selected topics include spin glasses, error correcting codes, satisfiability, and are central to each field. The approach focuses on large random instances, adopting a common probabilistic formulation in terms of graphical models. It presents message passing algorithms like belief propagation and survey propagation, and their use in decoding and constraint satisfaction solving. It also explains analysis techniques like density evolution and the cavity method, and uses them to study phase transitions.
Product Details
| ISBN-13: | 9780198570837 |
|---|---|
| Publisher: | Oxford University Press, USA |
| Publication date: | 03/27/2009 |
| Series: | Oxford Graduate Texts Series |
| Edition description: | New Edition |
| Pages: | 584 |
| Product dimensions: | 7.00(w) x 9.60(h) x 4.30(d) |
About the Author
Professor Marc Mezard
CNRS Research Director at Université de Paris Sud and Professor at Ecole Polytechnique, France
Marc Mezard received his PhD in 1984. He was hired in CNRS in 1981 and became research director in 1990 at Ecole Normale Supérieure. He joined the Université Paris Sud in 2001. He spent extensive periods in Rome University, in the KITP (Santa Barbara) and in MSRI (Berkeley). Author of about 150 publications, he has been awarded the silver medal of CNRS in 1990 and the Ampere price of the French academy of science in 1996. Dr Andrea Montanari
Assistant Professor, Stanford University and CNRS France
Andrea Montanari received a Laurea degree in Physics in 1997, and a Ph. D. in Theoretical Physics in 2001 (both from Scuola Normale Superiore in Pisa, Italy). He has been post-doctoral fellow at Laboratoire de Physique Théorique de l'Ecole Normale Supérieure (LPTENS), Paris, France, and the Mathematical Sciences Research Institute, Berkeley, USA. Since 2002 he is Chargé de Recherche (a permanent research position with Centre National de la Recherche Scientifique, CNRS) at LPTENS.
In September 2006 he joined Stanford University as Assistant Professor in the Departments of Electrical Engineering and Statistics.
In 2006 he was awarded the CNRS bronze medal for theoretical physics.
Table of Contents
Part II Independence
5 The random energy model 93
5.1 Definition of the model 93
5.2 Thermodynamics of the REM 94
5.3 The condensation phenomenon 100
5.4 A comment on quenched and annealed averages 101
5.5 The random subcube model 103
Notes 105
6 The random code ensemble 107
6.1 Code ensembles 107
6.2 The geometry of the random code ensemble 110
6.3 Communicating over a binary symmetric channel 112
6.4 Error-free communication with random codes 120
6.5 Geometry again: Sphere packing 123
6.6 Other random codes 126
6.7 A remark on coding theory and disordered systems 127
6.8 Appendix: Proof of Lemma 6.2 128
Notes 128
7 Number partitioning 131
7.1 A fair distribution into two groups? 131
7.2 Algorithmic issues 132
7.3 Partition of a random list: Experiments 133
7.4 The random cost model 136
7.5 Partition of a random list: Rigorous results 140
Notes 143
8 Introduction to replica theory 145
8.1 Replica solution of the random energy model 145
8.2 The fully connected p-spin glass model 155
8.3 Extreme value statistics and the REM 163
8.4 Appendix: Stability of the RS saddle point 166
Notes 169
Part III Models on Graphs
9 Factor graphs and graph ensembles 173
9.1 Factor graphs 173
9.2 Ensembles of factor graphs: Definitions 180
9.3 Random factor graphs: Basic properties 182
9.4 Random factor graphs: The giant component 187
9.5 The locally tree-like structure of random graphs 191
Notes 194
10 Satisfiability 197
10.1 The satisfiability problem 197
10.2 Algorithms 199
10.3 Random K-satisfiability ensembles 206
10.4 Random 2-SAT 209
10.5 The phase transition in random K(>q; 3)-SAT209
Notes 217
11 Low-density parity-check codes 219
11.1 Definitions 220
11.2 The geometry of the codebook 222
11.3 LDPC codes for the binary symmetric channel 231
11.4 A simple decoder: Bit flipping 236
Notes 239
12 Spin glasses 241
12.1 Spin glasses and factor graphs 241
12.2 Spin glasses: Constraints and frustration 245
12.3 What is a glass phase? 250
12.4 An example: The phase diagram of the SK model 262
Notes 265
13 Bridges: Inference and the Monte Carlo method 267
13.1 Statistical inference 268
13.2 The Monte Carlo method: Inference via sampling 272
13.3 Free-energy barriers 281
Notes 287
Part IV Short-Range Correlations
14 Belief propagation 291
14.1 Two examples 292
14.2 Belief propagation on tree graphs 296
14.3 Optimization: Max-product and min-sum 305
14.4 Loopy BP 310
14.5 General message-passing algorithms 316
14.6 Probabilistic analysis 317
Notes 325
15 Decoding with belief propagation 327
15.1 BP decoding: The algorithm 327
15.2 Analysis: Density evoluation 329
15.3 BP decoding for an erasure channel 342
15.4 The Bethe free energy and MAP decoding 347
Notes 352
16 The assignment problem 355
16.1 The assignment problem and random assignment ensembles 356
16.2 Message passing and its probabilistic analysis 357
16.3 A polynomial message-passing algorithm 366
16.4 Combinatorial results 371
16.5 An exercise: Multi-index assignment 376
Notes 378
17 Ising models on random graphs 381
17.1 The BP equations for Ising spins 381
17.2 RS cavity analysis 384
17.3 Ferromagnetic model 386
17.4 Spin glass models 391
Notes 399
Part V Long-Range Correlations
18 Linear equations with Boolean variables 403
18.1 Definitions and general remarks 404
18.2 Belief propagation 409
18.3 Core percolation and BP 412
18.4 The Sat-Unsat threshold in random Xorsat 415
18.5 The Hard-Sat phase: Clusters of solutions 421
18.6 An alternative approach: The cavity method 422
Notes 427
19 The 1RSB cavity method 429
19.1 Beyond BP: Many states 430
19.2 The 1RSB cavity equations 434
19.3 A first application: Xorsat 444
19.4 The special value x=1 449
19.5 Survey propagation 453
19.6 The nature of 1RSB phases 459
19.7 Appendix: The SP(y) equations for Xorsat 463
Notes 465
20 Random K-satisfiability 467
20.1 Belief propagation and the replica-symmetric analysis 468
20.2 Survey propagation and the 1RSB phase 474
20.3 Some ideas about the full phase diagram 485
20.4 An exercise: Colouring random graphs 488
Notes 491
21 Glassy states in coding theory 493
21.1 Local search algorithms and metastable states 493
21.2 The binary erasure channel 500
21.3 General binary memoryless symmetric channels 506
21.4 Metastable states and near-codewords 513
Notes 515
22 An ongoing story 517
22.1 Gibbs measures and long-range correlations 518
22.2 Higher levels of replica symmetry breaking 524
22.3 Phase structure and the behaviour of algorithms 535
Notes 538
Appendix A Symbols and notation 541
A.1 Equivalence relations 541
A.2 Orders of growth 542
A.3 Combinatorics and probability 543
A.4 Summary of mathematical notation 544
A.5 Information theory 545
A.6 Factor graphs 545
A.7 Cavity and message-passing methods 545
References 547
Index 565