Mathematical Aspects Of Spin Glasses And Neural Networks

This book aims to describe in simple terms the new area of statistical mechanics known as spin-glasses, encompassing systems in which quenched disorder is the dominant factor. The book begins with a non-mathematical explanation of the problem, and the modern understanding of the physics of the spin-glass state is formulated in general terms. Next, the 'magic' of the replica symmetry breaking scheme is demonstrated and the physics behind it discussed. Recent experiments on real spin-glass materials are briefly described to demonstrate how this somewhat abstract physics can be studied in the laboratory. The final chapters of the book are devoted to statistical models of neural networks. The material here is self-contained and should be accessible to students with a basic knowledge of theoretical physics and statistical mechanics. It has been used for a one-term graduate lecture course at the Landau Institute for Theoretical Physics. Contents:The Ising Magnetic SystemsPhysics of the Spin Glass StateReplica MethodReplica Symmetry BreakingPhysics of Replica Symmetry BreakingReplica Symmetry Breaking Solution Near TcUltrametricityScaling in the Space of Spin Glass StatesExperimentsPartial AnnealingStatistical Models of Neural NetworksThe Hopfield ModelPartial Annealing in Neural NetworksOther Kinds of Neural NetworksAppendix: Stability of the Replica-Symmetric Solutions Readership: Researchers and graduate students in statistical mechanics and neural networks. keywords: “The book by Viktor Dotsenko in large parts presents the most important results of this research based on the replica method. Although these results have been presented systematically already elsewhere (for instance in the well-known book by Amit) their concise presentation makes the book self-contained and a good introduction to the theoretical tools.” Mathematics Abstracts

Spin glasses are disordered magnetic systems that have led to the development of mathematical tools with an array of real-world applications, from airline scheduling to neural networks. Spin Glasses and Complexity offers the most concise, engaging, and accessible introduction to the subject, fully explaining what spin glasses are, why they are important, and how they are opening up new ways of thinking about complexity. This one-of-a-kind guide to spin glasses begins by explaining the fundamentals of order and symmetry in condensed matter physics and how spin glasses fit into--and modify--this framework. It then explores how spin-glass concepts and ideas have found applications in areas as diverse as computational complexity, biological and artificial neural networks, protein folding, immune response maturation, combinatorial optimization, and social network modeling. Providing an essential overview of the history, science, and growing significance of this exciting field, Spin Glasses and Complexity also features a forward-looking discussion of what spin glasses may teach us in the future about complex systems. This is a must-have book for students and practitioners in the natural and social sciences, with new material even for the experts.

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, that physicists studied by non-rigorous methods. They predicted spectacular behaviors, previously unknown in probability theory. They believe these behaviors occur in many models of considerable interest for several branches of science (statistical physics, neural networks and computer science). This book introduces in a rigorous manner this exciting new area to the mathematically minded reader. It requires no knowledge whatsoever of any physics, and contains proofs in complete detail of much of what is rigorously known on spin glasses at the time of writing.

The long list of “spin glass materials” and the summary of the experimental results provided in this book emphasize the common features of spin glasses despite the diversities. The critical review of more than a thousand papers not only identifies the complexities involved in the theoretical understanding of the static and dynamic properties of real spin glasses but also explains the physical concepts and mathematical formalism which have been used so successfully in solving the infinite range model. Morever, a beginner will find practical applications of the concepts of broken ergodicity, ultrametricity, gauge invariance, etc. in this book. Major progress has been made in solving many other challenging problems, e.g., computer design, associative memory, pattern recognition and neural networks, evolution of biological species etc. by mapping them onto the spin glass models. The chapter on these spin- glass-like systems will be useful not only to physicists but also to computer scientists and biologists. Contents:Real Spin Glass Materials, Spin-Glass-Like Systems and Spin Glass ModelsMean Field Theory of Spin GlassesSherrington-Kirkpatrick (SK) ModelVarious Solutions of the SK Model and Their EquivalenceDynamics of Spin GlassesVector Spin Glasses and Mixed PhaseResults of Computer ExperimentsRandom Energy Model Readership: Condensed matter physicists and material scientists. Keywords:Spin Glass Materials;Ergodicity;Ultrametricity;Gauge Invariance;Sherrington-Kirkpatrick (SK) Model;Random Energy Model

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.

This book serves as a concise introduction to the state-of-the-art of spin glass theory. The collection of review papers are written by leading experts in the field and cover the topic from a wide variety of angles. The book will be useful to both graduate students and young researchers, as well as to anyone curious to know what is going on in this exciting area of mathematical physics.