Designed for readers with an undergraduate mathematics background, this introduction offers a vibrant and scholarly exploration of physics concepts. It bridges the gap between math and physics, making complex ideas accessible to those new to the subject. The book emphasizes clarity and engagement, ensuring that readers can grasp fundamental principles while appreciating the beauty of physics.
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.
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.
Cavity and Mean Field Models
The book explores the groundbreaking work of theoretical physicists in the 1980s on spin glasses, which are disordered systems characterized by unique random structures. It highlights the non-rigorous methods used to study these models and the surprising behaviors they predicted, expanding the understanding of probability theory. The implications of these findings extend across various scientific fields, including statistical physics, neural networks, and computer science, showcasing their relevance and potential impact on future research.
The book presents a comprehensive and revised exploration of spin glasses, addressing the challenges they pose for mathematicians. This first volume focuses on fundamental results and methodologies, laying the groundwork for understanding this complex topic. An anticipated second volume is set to follow, promising to expand on these concepts further.
Modern Methods and Classical Problems
Focusing on generic chaining, the book explores its role in achieving optimal bounds for Gaussian and Bernoulli processes. It delves into modern stochastic methods, highlighting significant applications and providing comprehensive solutions to classical problems, making it a valuable resource for those interested in advanced statistical techniques.
Volume II: Advanced Replica-Symmetry and Low Temperature
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.