This book provides an in-depth account of modern methods used to bound the supremum of stochastic processes. Starting from first principles, it takes the reader to the frontier of current research. This second edition has been completely rewritten, offering substantial improvements to the exposition and simplified proofs, as well as new results. The book starts with a thorough account of the generic chaining, a remarkably simple and powerful method to bound a stochastic process that should belong to every probabilist's toolkit. The effectiveness of the scheme is demonstrated by the characterization of sample boundedness of Gaussian processes. Much of the book is devoted to exploring the wealth of ideas and results generated by thirty years of efforts to extend this result to more general classes of processes, culminating in the recent solution of several key conjectures. A large part of this unique book is devoted to the author's influential work. While many of the results presented are rather advanced, others bear on the very foundations of probability theory. In addition to providing an invaluable reference for researchers, the book should therefore also be of interest to a wide range of readers.

This is a new, completely revised, updated and enlarged edition of the author's Ergebnisse vol. 46: "Spin Glasses: A Challenge for Mathematicians" in two volumes (this is the 2nd volume). 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 present Volume II contains a considerable amount of new material, in particular all the fundamental low-temperature results obtained after the publication of the first edition.

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 fundamental question of characterizing continuity and boundedness of Gaussian processes goes back to Kolmogorov. After contributions by R. Dudley and X. Fernique, it was solved by the author. This book provides an overview of "generic chaining," a completely natural variation on the ideas of Kolmogorov. It takes the reader from the first principles to the edge of current knowledge and to the open problems that remain in this domain.

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.

Quantum field theory (QFT) is one of the great achievements of physics, of profound interest to mathematicians. Most pedagogical texts on QFT are geared toward budding professional physicists, however, whereas mathematical accounts are abstract and difficult to relate to the physics. This book bridges the gap. While the treatment is rigorous whenever possible, the accent is not on formality but on explaining what the physicists do and why, using precise mathematical language. In particular, it covers in detail the mysterious procedure of renormalization. Written for readers with a mathematical background but no previous knowledge of physics and largely self-contained, it presents both basic physical ideas from special relativity and quantum mechanics and advanced mathematical concepts in complete detail. It will be of interest to mathematicians wanting to learn about QFT and, with nearly 300 exercises, also to physics students seeking greater rigor than they typically find in their courses.

The book develops modern methods and in particular the "generic chaining" to bound stochastic processes. This methods allows in particular to get optimal bounds for Gaussian and Bernoulli processes. Applications are given to stable processes, infinitely divisible processes, matching theorems, the convergence of random Fourier series, of orthogonal series, and to functional analysis. The complete solution of a number of classical problems is given in complete detail, and an ambitious program for future research is laid out.

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.

New isoperimetric inequalities and random process techniques have recently appeared at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (e.g. boundedness and continuity of random processes, integrability and limit theorems for vector valued random variables) and of some of their links to Geometry of Banach spaces. Its purpose is to present some of the main aspects of this theory, from the foundations to the latest developments, treated with the most recent and updated tools. In particular, the most important features are the systematic use of isoperimetry and related concentration of measure phenomena (to study integrability and limit theorems for vector valued random variables), and recent abstract random process techniques (entropy and majorizing measures). Some examples of these probabilistic ideas to classical Banach space theory complete this exposition.

The fundamental question of characterizing continuity and boundedness of Gaussian processes goes back to Kolmogorov. After contributions by R. Dudley and X. Fernique, it was solved by the author. This book provides an overview of "generic chaining," a completely natural variation on the ideas of Kolmogorov. It takes the reader from the first principles to the edge of current knowledge and to the open problems that remain in this domain.

This is a new, completely revised, updated and enlarged edition of the author's Ergebnisse vol. 46: "Spin Glasses: A Challenge for Mathematicians" in two volumes (this is the 2nd volume). 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 present Volume II contains a considerable amount of new material, in particular all the fundamental low-temperature results obtained after the publication of the first edition.