Directed Polymers in Random Environments
Directed Polymers in Random Environments

Author: Francis Comets

Publisher: Springer

Published: 2017-01-26

Total Pages: 210

ISBN-13: 3319504878

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Analyzing the phase transition from diffusive to localized behavior in a model of directed polymers in a random environment, this volume places particular emphasis on the localization phenomenon. The main questionis: What does the path of a random walk look like if rewards and penalties are spatially randomly distributed?This model, which provides a simplified version of stretched elastic chains pinned by random impurities, has attracted much research activity, but it (and its relatives) still holds many secrets, especially in high dimensions. It has non-gaussian scaling limits and it belongs to the so-called KPZ universality class when the space is one-dimensional. Adopting a Gibbsian approach, using general and powerful tools from probability theory, the discrete model is studied in full generality. Presenting the state-of-the art from different perspectives, and written in the form of a first course on the subject, this monograph is aimed at researchers in probability or statistical physics, but is also accessible to masters and Ph.D. students.

Correlated Random Systems: Five Different Methods
Correlated Random Systems: Five Different Methods

Author: Véronique Gayrard

Publisher: Springer

Published: 2015-06-09

Total Pages: 213

ISBN-13: 3319176749

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This volume presents five different methods recently developed to tackle the large scale behavior of highly correlated random systems, such as spin glasses, random polymers, local times and loop soups and random matrices. These methods, presented in a series of lectures delivered within the Jean-Morlet initiative (Spring 2013), play a fundamental role in the current development of probability theory and statistical mechanics. The lectures were: Random Polymers by E. Bolthausen, Spontaneous Replica Symmetry Breaking and Interpolation Methods by F. Guerra, Derrida's Random Energy Models by N. Kistler, Isomorphism Theorems by J. Rosen and Spectral Properties of Wigner Matrices by B. Schlein. This book is the first in a co-edition between the Jean-Morlet Chair at CIRM and the Springer Lecture Notes in Mathematics which aims to collect together courses and lectures on cutting-edge subjects given during the term of the Jean-Morlet Chair, as well as new material produced in its wake. It is targeted at researchers, in particular PhD students and postdocs, working in probability theory and statistical physics.

Probability and Analysis in Interacting Physical Systems
Probability and Analysis in Interacting Physical Systems

Author: Peter Friz

Publisher: Springer

Published: 2019-05-24

Total Pages: 303

ISBN-13: 303015338X

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This Festschrift on the occasion of the 75th birthday of S.R.S. Varadhan, one of the most influential researchers in probability of the last fifty years, grew out of a workshop held at the Technical University of Berlin, 15–19 August, 2016. This volume contains ten research articles authored by several of Varadhan's former PhD students or close collaborators. The topics of the contributions are more or less closely linked with some of Varadhan's deepest interests over the decades: large deviations, Markov processes, interacting particle systems, motions in random media and homogenization, reaction-diffusion equations, and directed last-passage percolation. The articles present original research on some of the most discussed current questions at the boundary between analysis and probability, with an impact on understanding phenomena in physics. This collection will be of great value to researchers with an interest in models of probability-based statistical mechanics.

Probability and Statistical Physics in Two and More Dimensions
Probability and Statistical Physics in Two and More Dimensions

Author: Clay Mathematics Institute. Summer School

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 481

ISBN-13: 0821868632

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This volume is a collection of lecture notes for six of the ten courses given in Buzios, Brazil by prominent probabilists at the 2010 Clay Mathematics Institute Summer School, ``Probability and Statistical Physics in Two and More Dimensions'' and at the XIV Brazilian School of Probability. In the past ten to fifteen years, various areas of probability theory related to statistical physics, disordered systems and combinatorics have undergone intensive development. A number of these developments deal with two-dimensional random structures at their critical points, and provide new tools and ways of coping with at least some of the limitations of Conformal Field Theory that had been so successfully developed in the theoretical physics community to understand phase transitions of two-dimensional systems. Included in this selection are detailed accounts of all three foundational courses presented at the Clay school--Schramm-Loewner Evolution and other Conformally Invariant Objects, Noise Sensitivity and Percolation, Scaling Limits of Random Trees and Planar Maps--together with contributions on Fractal and Multifractal properties of SLE and Conformal Invariance of Lattice Models. Finally, the volume concludes with extended articles based on the courses on Random Polymers and Self-Avoiding Walks given at the Brazilian School of Probability during the final week of the school. Together, these notes provide a panoramic, state-of-the-art view of probability theory areas related to statistical physics, disordered systems and combinatorics. Like the lectures themselves, they are oriented towards advanced students and postdocs, but experts should also find much of interest.

Random Polymers
Random Polymers

Author: Frank Hollander

Publisher: Springer Science & Business Media

Published: 2009-05-14

Total Pages: 271

ISBN-13: 364200332X

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Polymer chains that interact with themselves and/or their environment display a range of physical and chemical phenomena. This text focuses on the mathematical description of some of these phenomena, offering a mathematical panorama of polymer chains.

Random Polymer Models
Random Polymer Models

Author: Giambattista Giacomin

Publisher: Imperial College Press

Published: 2007

Total Pages: 259

ISBN-13: 1860948294

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Random polymer models and their applications -- The homogeneous pinning model -- Weakly inhomogeneous models -- The free energy of disordered polymer chains -- Disordered pinning models: The hase diagram -- Disordered copolymers and selective interfaces: The phase diagram -- The localized phase of disordered polymers -- The delocalized phase of disordered polymers -- Numerical algorithms and computations

Brownian Motion, Obstacles and Random Media
Brownian Motion, Obstacles and Random Media

Author: Alain-Sol Sznitman

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 366

ISBN-13: 3662112817

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This book provides an account for the non-specialist of the circle of ideas, results and techniques, which grew out in the study of Brownian motion and random obstacles. It also includes an overview of known results and connections with other areas of random media, taking a highly original and personal approach throughout.

Probability in Complex Physical Systems
Probability in Complex Physical Systems

Author: Jean-Dominique Deuschel

Publisher: Springer Science & Business Media

Published: 2012-04-23

Total Pages: 518

ISBN-13: 3642238114

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Probabilistic approaches have played a prominent role in the study of complex physical systems for more than thirty years. This volume collects twenty articles on various topics in this field, including self-interacting random walks and polymer models in random and non-random environments, branching processes, Parisi formulas and metastability in spin glasses, and hydrodynamic limits for gradient Gibbs models. The majority of these articles contain original results at the forefront of contemporary research; some of them include review aspects and summarize the state-of-the-art on topical issues – one focal point is the parabolic Anderson model, which is considered with various novel aspects including moving catalysts, acceleration and deceleration and fron propagation, for both time-dependent and time-independent potentials. The authors are among the world’s leading experts. This Festschrift honours two eminent researchers, Erwin Bolthausen and Jürgen Gärtner, whose scientific work has profoundly influenced the field and all of the present contributions.

The Parabolic Anderson Model
The Parabolic Anderson Model

Author: Wolfgang König

Publisher: Birkhäuser

Published: 2016-06-30

Total Pages: 199

ISBN-13: 3319335960

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This is a comprehensive survey on the research on the parabolic Anderson model – the heat equation with random potential or the random walk in random potential – of the years 1990 – 2015. The investigation of this model requires a combination of tools from probability (large deviations, extreme-value theory, e.g.) and analysis (spectral theory for the Laplace operator with potential, variational analysis, e.g.). We explain the background, the applications, the questions and the connections with other models and formulate the most relevant results on the long-time behavior of the solution, like quenched and annealed asymptotics for the total mass, intermittency, confinement and concentration properties and mass flow. Furthermore, we explain the most successful proof methods and give a list of open research problems. Proofs are not detailed, but concisely outlined and commented; the formulations of some theorems are slightly simplified for better comprehension.

On Dobrushin's Way. From Probability Theory to Statistical Physics
On Dobrushin's Way. From Probability Theory to Statistical Physics

Author: Robert A. Minlos

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 260

ISBN-13: 9780821821503

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Fellow Russian mathematicians discuss and extend the works of Dobrushin (1929-95,), who worked in many areas of mathematics, but had deepest influence on mathematical physics and was one of the founders of the rigorous study of statistical physics. The 15 technical papers are flanked by a short biography and recollections by colleagues and students. The topics include the lower spectral branch of the generator of the stochastic dynamics for the classical Heisenberg model, non-symmetric simple random walks along orbits of ergodic automorphisms, the Cramer transform and large deviations on three- dimensional Lobachevsky space, and dynamics of Ising-spin systems at zero temperature. No index is provided. Annotation copyrighted by Book News, Inc., Portland, OR.