Gaussian Processes on Trees: From Spin Glasses to Branching Brownian Motion (Cambridge Studies in Advanced Mathematics Book 163)
Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a concise review of classical extreme value statistics and a basic introduction to mean-field spin glasses, the author then focuses on branching Brownian motion. Here, the classical results of Bramson on the asymptotics of solutions of the F-KPP equation are reviewed in detail and applied to the recent construction of the extremal process of BBM. The extension of these results to branching Brownian motion with variable speed are then explained. As a self-contained exposition that is accessible to graduate students with some background in probability theory, this book makes a good introduction for anyone interested in accessing this exciting field of mathematics.
Published by: Cambridge University Press | Publication date: 10/20/2016Kindle book details: Kindle Edition, 211 pages
Metastability: A Potential-Theoretic Approach (Grundlehren der mathematischen Wissenschaften Book 351)
This monograph provides a concise presentation of a mathematical approach to metastability, a wide-spread phenomenon in the dynamics of non-linear systems - physical, chemical, biological or economic - subject to the action of temporal random forces typically referred to as noise, based on potential theory of reversible Markov processes. The authors shed new light on the metastability phenomenon as a sequence of visits of the path of the process to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets.The theory is illustrated with many examples, ranging from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata, unveiling the common universal features of these systems with respect to their metastable behaviour. The monograph will serve both as comprehensive introduction and as reference for graduate students and researchers interested in metastability.
Published by: Springer | Publication date: 03/14/2016Kindle book details: Kindle Edition, 581 pages
Focusing on the mathematics that lies at the intersection of probability theory, statistical physics, combinatorics and computer science, this volume collects together lecture notes on recent developments in the area. The common ground of these subjects is perhaps best described by the three terms in the title: Random Walks, Random Fields and Disordered Systems. The specific topics covered include a study of Branching Brownian Motion from the perspective of disordered (spin-glass) systems, a detailed analysis of weakly self-avoiding random walks in four spatial dimensions via methods of field theory and the renormalization group, a study of phase transitions in disordered discrete structures using a rigorous version of the cavity method, a survey of recent work on interacting polymers in the ballisticity regime and, finally, a treatise on two-dimensional loop-soup models and their connection to conformally invariant systems and the Gaussian Free Field. The notes are aimed at early graduate students with a modest background in probability and mathematical physics, although they could also be enjoyed by seasoned researchers interested in learning about recent advances in the above fields.
Published by: Springer | Publication date: 09/21/2015Kindle book details: Kindle Edition, 254 pages
Statistical Mechanics of Disordered Systems: A Mathematical Perspective (Cambridge Series in Statistical and Probabilistic Mathematics Book 18)
This self-contained book is a graduate-level introduction for mathematicians and for physicists interested in the mathematical foundations of the field, and can be used as a textbook for a two-semester course on mathematical statistical mechanics. It assumes only basic knowledge of classical physics and, on the mathematics side, a good working knowledge of graduate-level probability theory. The book starts with a concise introduction to statistical mechanics, proceeds to disordered lattice spin systems, and concludes with a presentation of the latest developments in the mathematical understanding of mean-field spin glass models. In particular, progress towards a rigorous understanding of the replica symmetry-breaking solutions of the Sherrington-Kirkpatrick spin glass models, due to Guerra, Aizenman-Sims-Starr and Talagrand, is reviewed in some detail.
Published by: Cambridge University Press | Publication date: 06/08/2006Kindle book details: Kindle Edition, 328 pages
This volume presents a collection of courses introducing the reader to the recent progress with attention being paid to laying solid grounds and developing various basic tools. It presents new results on phase transitions for gradient lattice models.
Published by: Springer | Publication date: 07/31/2009Kindle book details: Kindle Edition, 360 pages
Aimed at graduates and potential researchers, this is a comprehensive introduction to the mathematical aspects of spin glasses and neural networks. It should be useful to mathematicians in probability theory and theoretical physics, and to engineers working in theoretical computer science.
Published by: Birkhäuser | Publication date: 02/25/2012Kindle book details: Kindle Edition, 382 pages
The proceedings of the 2005 les Houches summer school on Mathematical Statistical Physics give and broad and clear overview on this fast developing area of interest to both physicists and mathematicians.
- Introduction to a field of math with many interdisciplinary connections in physics, biology, and computer science
- Roadmap to the next decade of mathematical statistical mechanics
- Volume for reference years to come
Published by: Elsevier Science | Publication date: 06/27/2006Kindle book details: Kindle Edition, 848 pages
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.
Published by: Springer | Publication date: 01/11/2007Kindle book details: Kindle Edition, 187 pages