Information, Physics and Computation, by Marc Mézard and Andrea Montanari

Almost without notice, a new field of research has emerged between applied mathematics and computational physics. This derives from a combination of algorithms and protocols to handle vast numbers of basic entities and a growing interest in treating models of random materials and heterogeneous substances with their interfaces. The unifying concept is statistical methods and their applications to vide ranges of problems. Its central concepts like entropy are given an essential role in the approach. The book considered here aims at collecting the various directions of this research into an autonomous topic where similarities of methods and relationships between concepts offer a unified view of seemingly very different endeavours. This goal is achievable because the common methods are geared to utilisation of the increasing powers of computers applied to complex problems. Mathematics has contributed classically hard computational tasks like the ‘travelling salesman’ problem. Also communication and message coding have offered many challenges, of which the factorisation of large numbers is the most well known. In physics, highly complex systems like polymer networks and protein folding have actuality in realworld contexts. The trailblazer turned out to be the spin glass system. Its model has been studied because it offers understanding of the behaviour of common glassy substances. Its many locally stable configurations contribute another computationally hard problem. The book considered presents problems of significant degree of hardness and complexity. Similarity of approach brings out common features. As an example one may mention the role of phase transitions or thresholds. It is, namely, found that the hardest-tosolve cases often occur near such configurations. The spin glass serves as an example where progress in one application carriages over into others. The numerical annealing methods developed for this system have become a standard tool for optimisation in other complex systems; its applications range from biological science to economic modelling. The present book is highly abstract. The presentation rests on formal definitions and their analyses, derivations and proofs. There are no physical arguments for the various concepts and models. They are simply taken over from their original settings. Likewise, the results are considered only as computational processes. The book proceeds according to increasing degree of complexity. It starts from basic knowledge of communication theory and statistical methods. These considerations are then applied to both communication systems and physical models. The replica approach emerged from the spin glass research. This method is presented in some detail. Methods based on graphs are introduced and explored. As part of logical analyses the concept of satisfiability plays an essential role. In solving a system of logical constraints it acts like solvability for equations. The authors make extensive use of a ‘cavity method’ where the results are evaluated by excluding one basic unit after another systematically. While presenting ever more sophisticated methods, the authors apply them to problems of message transmission, information coding and models of random materials. They offer a rich selection of tools, models and solutions. The literature is covered by many references to original works. They conclude their presentation by an exposition of remaining challenges and visions of things to come. I have no doubts that this book is very valuable for someone wishing to take up serious research in this lively and exciting new field. However clear and concise the presentation is, the material is highly abstract and extremely technical. Consequently, the book will appeal only to those who aim at original research in their area. For other potential readers, the material is hard to digest. The book is a highly valuable collection of material, but its readership is destined to be strictly limited.