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.

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.

This superb new book is one of the first publications in recent years to provide a broad overview of this interdisciplinary field. Most of the book is written in a self contained manner, assuming only a general knowledge of statistical mechanics and basic probabilty theory . It provides the reader with a sound introduction to the field and to the analytical techniques necessary to follow its most recent developments

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.

Over the last decade, spin glass theory has turned from a fascinating part of t- oretical physics to a ?ourishing and rapidly growing subject of probability theory as well. These developments have been triggered to a large part by the mathem- ical understanding gained on the fascinating and previously mysterious “Parisi solution” of the Sherrington–Kirkpatrick mean ?eld model of spin glasses, due to the work of Guerra, Talagrand, and others. At the same time, new aspects and applications of the methods developed there have come up. The presentvolumecollects a number of reviewsaswellas shorterarticlesby lecturers at a summer school on spin glasses that was held in July 2007 in Paris. These articles range from pedagogical introductions to state of the art papers, covering the latest developments. In their whole, they give a nice overview on the current state of the ?eld from the mathematical side. The review by Bovier and Kurkova gives a concise introduction to mean ?eld models, starting with the Curie–Weiss model and moving over the Random Energymodels up to the Parisisolutionof the Sherrington–Kirkpatrikmodel. Ben Arous and Kuptsov present a more recent view and disordered systems through the so-called local energy statistics. They emphasize that there are many ways to look at Hamiltonians of disordered systems that make appear the Random Energy model (or independent random variables) as a universal mechanism for describing certain rare events. An important tool in the analysis of spin glasses are correlation identities.

This volume is an introduction to the application of techniques developed for the study of disordered systems to problems which arise in biology. Topics presented include neural networks, adaptation and evolution, maturation of the immune response, and protein dynamics and folding. This book will appeal to students and researchers interested in statistical and condensed matter physics, glasses and spin glasses, and biophysics.

The celebrated Parisi solution of the Sherrington-Kirkpatrick model for spin glasses is one of the most important achievements in the field of disordered systems. Over the last three decades, through the efforts of theoretical physicists and mathematicians, the essential aspects of the Parisi solution were clarified and proved mathematically. The core ideas of the theory that emerged are the subject of this book, including the recent solution of the Parisi ultrametricity conjecture and a conceptually simple proof of the Parisi formula for the free energy. The treatment is self-contained and should be accessible to graduate students with a background in probability theory, with no prior knowledge of spin glasses. The methods involved in the analysis of the Sherrington-Kirkpatrick model also serve as a good illustration of such classical topics in probability as the Gaussian interpolation and concentration of measure, Poisson processes, and representation results for exchangeable arrays.