Anton Bovier

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.

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.

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.

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.

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.

The Lecture Notes collect seven mini-courses presented at the 5th Prague Summer School on Mathematical Statistical Physics that took placeduringtwoweeksofSeptember2006.Aswithprecedingschools,it was aimed at PhD students and young postdocs. The central theme of the volume is what could be called "mathematics of phase transitions" in diverse contexts. Even though all courses were meant to introduce the reader to recent progress of a particular topic of modern statis- cal physics, attention has been paid to providing a solid grounding by carefully developing various basic tools. One of the techniques that led, more than two decades ago, to a seriesofimportantoutcomesinthetheoryofphasetransitionsoflattice models was re?ection positivity. Recently it resurfaced and was used to obtain interesting new results in various settings. The lectures of Marek Biskup include a thorough introduction to re?ection positivity as well as a review of its recent applications. In addition, it contains a crash course on lattice spin models that is useful as a background for other lectures of the collection. Also the following two contributions concern equilibrium statistical physics.ThelecturesofDmitriIo?earedevotedtoastochasticgeom- ricreformulationofclassicalaswellasquantumIsingmodels. Auni?ed approachtotheFortuin-Kasteleynandrandomcurrentrepresentations in terms of path integrals is presented. Statistical mechanics of directed polymers interacting with o- dimensionalspatiale?ectsisatopicwithvariousapplicationsinphysics and biophysics. The lectures of Fabio Toninelli are devoted to a th- ough discussion of the localization/delocalization transition in these models.

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.

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.