Download or read book Conformal Invariance written by and published by Springer. This book was released on 2012-04-06 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Conformal Invariance an Introduction to Loops Interfaces and Stochastic Loewner Evolution written by Malte Henkel and published by Springer Science & Business Media. This book was released on 2012-04-04 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Conformal invariance has been a spectacularly successful tool in advancing our understanding of the two-dimensional phase transitions found in classical systems at equilibrium. This volume sharpens our picture of the applications of conformal invariance, introducing non-local observables such as loops and interfaces before explaining how they arise in specific physical contexts. It then shows how to use conformal invariance to determine their properties. Moving on to cover key conceptual developments in conformal invariance, the book devotes much of its space to stochastic Loewner evolution (SLE), detailing SLE’s conceptual foundations as well as extensive numerical tests. The chapters then elucidate SLE’s use in geometric phase transitions such as percolation or polymer systems, paying particular attention to surface effects. As clear and accessible as it is authoritative, this publication is as suitable for non-specialist readers and graduate students alike.
Download or read book Conformal Invariance an Introduction to Loops Interfaces and Stochastic Loewner Evolution written by Malte Henkel and published by Springer Science & Business Media. This book was released on 2012-04-05 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Conformal invariance has been a spectacularly successful tool in advancing our understanding of the two-dimensional phase transitions found in classical systems at equilibrium. This volume sharpens our picture of the applications of conformal invariance, introducing non-local observables such as loops and interfaces before explaining how they arise in specific physical contexts. It then shows how to use conformal invariance to determine their properties. Moving on to cover key conceptual developments in conformal invariance, the book devotes much of its space to stochastic Loewner evolution (SLE), detailing SLE’s conceptual foundations as well as extensive numerical tests. The chapters then elucidate SLE’s use in geometric phase transitions such as percolation or polymer systems, paying particular attention to surface effects. As clear and accessible as it is authoritative, this publication is as suitable for non-specialist readers and graduate students alike.
Download or read book Advances in Disordered Systems Random Processes and Some Applications written by Pierluigi Contucci and published by Cambridge University Press. This book was released on 2017 with total page 383 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a unified perspective on the study of complex systems with contributions written by leading scientists from various disciplines, including mathematics, physics, computer science, biology, economics and social science. It is written for researchers from a broad range of scientific fields with an interest in recent developments in complex systems.
Download or read book Conformal Loop Ensembles and the Gaussian Free Field written by Samuel Stewart Watson and published by . This book was released on 2015 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of two-dimensional statistical physics models leads naturally to the analysis of various conformally invariant mathematical objects, such as the Gaussian free field, the Schramm-Loewner evolution, and the conformal loop ensemble. Just as Brownian motion is a scaling limit of discrete random walks, these objects serve as universal scaling limits of functions or paths associated with the underlying discrete models. We establish a new convergence result for percolation, a well-studied discrete model. We also study random sets of points surrounded by exceptional numbers of conformal loop ensemble loops and establish the existence of a random generalized function describing the nesting of the conformal loop ensemble. Using this framework, we study the relationship between Gaussian free field extrema and nesting extrema of the ensemble of Gaussian free field level loops. Finally, we describe a coupling between the set of all Gaussian free field level loops and a conformal loop ensemble growth process introduced by Werner and Wu. We prove that the dynamics are determined by the conformal loop ensemble in this coupling, and we use this result to construct a conformally invariant metric space.
Download or read book Probability Geometry and Integrable Systems written by Mark Pinsky and published by Cambridge University Press. This book was released on 2008-03-17 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reflects the range of mathematical interests of Henry McKean, to whom it is dedicated.
Download or read book Random Walks and Geometry written by Vadim Kaimanovich and published by Walter de Gruyter. This book was released on 2008-08-22 with total page 545 pages. Available in PDF, EPUB and Kindle. Book excerpt: Die jüngsten Entwicklungen zeigen, dass sich Wahrscheinlichkeitsverfahren zu einem sehr wirkungsvollen Werkzeug entwickelt haben, und das auf so unterschiedlichen Gebieten wie statistische Physik, dynamische Systeme, Riemann'sche Geometrie, Gruppentheorie, harmonische Analyse, Graphentheorie und Informatik.
Download or read book Schramm Loewner Evolution written by Antti Kemppainen and published by Springer. This book was released on 2017-12-22 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a short, but complete, introduction to the Loewner equation and the SLEs, which are a family of random fractal curves, as well as the relevant background in probability and complex analysis. The connection to statistical physics is also developed in the text in an example case. The book is based on a course (with the same title) lectured by the author. First three chapters are devoted to the background material, but at the same time, give the reader a good understanding on the overview on the subject and on some aspects of conformal invariance. The chapter on the Loewner equation develops in detail the connection of growing hulls and the differential equation satisfied by families of conformal maps. The Schramm–Loewner evolutions are defined and their basic properties are studied in the following chapter, and the regularity properties of random curves as well as scaling limits of discrete random curves are investigated in the final chapter. The book is aimed at graduate students or researchers who want to learn the subject fairly quickly.
Download or read book Probability and Statistical Physics in Two and More Dimensions written by Clay Mathematics Institute. Summer School and published by American Mathematical Soc.. This book was released on 2012 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Selected Works of Oded Schramm written by Itai Benjamini and published by Springer Science & Business Media. This book was released on 2011-08-12 with total page 1199 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is dedicated to the memory of the late Oded Schramm (1961-2008), distinguished mathematician. Throughout his career, Schramm made profound and beautiful contributions to mathematics that will have a lasting influence. In these two volumes, Editors Itai Benjamini and Olle Häggström have collected some of his papers, supplemented with three survey papers by Steffen Rohde, Häggström and Cristophe Garban that further elucidate his work. The papers within are a representative collection that shows the breadth, depth, enthusiasm and clarity of his work, with sections on Geometry, Noise Sensitivity, Random Walks and Graph Limits, Percolation, and finally Schramm-Loewner Evolution. An introduction by the Editors and a comprehensive bibliography of Schramm's publications complete the volume. The book will be of especial interest to researchers in probability and geometry, and in the history of these subjects.
Download or read book Conformally Invariant Processes in the Plane written by Gregory F. Lawler and published by American Mathematical Soc.. This book was released on 2008 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents an introduction to the conformally invariant processes that appear as scaling limits. This book covers such topics as stochastic integration, and complex Brownian motion and measures derived from Brownian motion. It is suitable for those interested in random processes and their applications in theoretical physics.
Download or read book An Introduction to Stochastic Loewner Evolution written by John D. Mangual and published by . This book was released on 2006 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Integrability in Random Conformal Geometry written by Jie Jun Ang and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Liouville quantum gravity (LQG) is a random surface arising as the scaling limit of random planar maps. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in many statistical physics models. Liouville conformal field theory (LCFT) is the quantum field theory underlying LQG. Each of these satisfies conformal invariance or covariance. This thesis proves exact formulas in random conformal geometry; we highlight a few here. The Brownian annulus describes the scaling limit of uniform random planar maps with the annulus topology, and is the canonical annular [gamma]-LQG surface with [gamma] = [square root]8/3. We obtain the law of its modulus, which is as predicted from the ghost partition function in bosonic string theory. The conformal loop ensemble (CLE) is a random collection of loops in the plane which locally look like SLE, corresponding to the scaling limit of all interfaces in several important statistical mechanics models. We derive the three-point nesting statistic of simple CLE on the sphere. It agrees with the imaginary DOZZ formula of Zamolodchikov (2005) and Kostov-Petkova (2007), which is the three-point structure constant of the generalized minimal model conformal field theories. We compute the one-point bulk structure constant for LCFT on the disk, thereby proving the formula proposed by Fateev, Zamolodchikov and Zamolodchikov (2000). This is a disk analog of the DOZZ constant for the sphere. Our result represents the first step towards solving LCFT on surfaces with boundary via the conformal bootstrap. Our arguments depend on the interplay between LQG, SLE and LCFT. Firstly, LQG behaves well under conformal welding with SLE curves as the interfaces. Secondly, LCFT and LQG give complementary descriptions of the same geometry.
Download or read book Markov Processes and Related Fields written by and published by . This book was released on 2007 with total page 848 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematical Reviews written by and published by . This book was released on 2007 with total page 804 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Probability on Graphs written by Geoffrey Grimmett and published by Cambridge University Press. This book was released on 2018-01-25 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. This new edition features accounts of major recent progress, including the exact value of the connective constant of the hexagonal lattice, and the critical point of the random-cluster model on the square lattice. The choice of topics is strongly motivated by modern applications, and focuses on areas that merit further research. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.
Download or read book The Random Cluster Model written by Geoffrey R. Grimmett and published by Springer Science & Business Media. This book was released on 2006-12-13 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: The random-cluster model has emerged as a key tool in the mathematical study of ferromagnetism. It may be viewed as an extension of percolation to include Ising and Potts models, and its analysis is a mix of arguments from probability and geometry. The Random-Cluster Model contains accounts of the subcritical and supercritical phases, together with clear statements of important open problems. The book includes treatment of the first-order (discontinuous) phase transition.
