Information, Physics, and Computation »

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.

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.

Pt. I Background

1 Introduction to information theory 3

2 Statistical physics and probability theory 23

3 Introduction to combinatorial optimization 47

4 A probabilistic toolbox 65

Pt. II Independence

5 The random energy model 93

6 The random code ensemble 107

7 Number partitioning 131

8 Introduction to replica theory 145

Pt. III Models on Graphs

9 Factor graphs and graph ensembles 173

10 Satisfiability 197

11 Low-density parity-check codes 219

12 Spin glasses 241

13 Bridges: Inference and the Monte Carlo method 267

Pt. IV Short-Range Correlations

14 Belief propagation 291

15 Decoding with belief propagation 327

16 The assignment problem 355

17 Ising models on random graphs 381

Pt. V Long-Range Correlations

18 Linear equations with Boolean variables 403

19 The 1RSB cavity method 429

20 Random K-satisfiability 467

21 Glassy states in coding theory 493

22 An ongoing story 517

App. A Symbols and notation 541

References 547

Index 565