The Critical Mean Field Random Cluster Model
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The Random-Cluster Model
Author | : Geoffrey R. Grimmett |
Publisher | : Springer Science & Business Media |
Total Pages | : 392 |
Release | : 2006-12-13 |
Genre | : Mathematics |
ISBN | : 3540328912 |
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.
The Random-Cluster Model
Author | : Geoffrey R. Grimmett |
Publisher | : Springer |
Total Pages | : 378 |
Release | : 2009-09-02 |
Genre | : Mathematics |
ISBN | : 9783540821588 |
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.
Probability on Graphs
Author | : Geoffrey Grimmett |
Publisher | : Cambridge University Press |
Total Pages | : 279 |
Release | : 2018-01-25 |
Genre | : Mathematics |
ISBN | : 1108542999 |
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.
A Power Law of Order 1/4 for Critical Mean Field Swendsen-Wang Dynamics
Author | : Yun Long |
Publisher | : American Mathematical Soc. |
Total Pages | : 96 |
Release | : 2014-09-29 |
Genre | : Mathematics |
ISBN | : 1470409100 |
Introduction Statement of the results Mixing time preliminaries Outline of the proof of Theorem 2.1 Random graph estimates Supercritical case Subcritical case Critical Case Fast mixing of the Swendsen-Wang process on trees Acknowledgements Bibliography
The Random-Cluster Model
Author | : Geoffrey Grimmett |
Publisher | : Springer Verlag |
Total Pages | : 377 |
Release | : 2006-01-01 |
Genre | : Mathematics |
ISBN | : 9783540328902 |
The random-cluster model has emerged in recent years 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. This systematic study includes accounts of the subcritical and supercritical phases, together with clear statements of important open problems. There is an extensive treatment of the first-order (discontinuous) phase transition, as well as a chapter devoted to applications of the random-cluster method to other models of statistical physics.
Random Graphs, Phase Transitions, and the Gaussian Free Field
Author | : Martin T. Barlow |
Publisher | : Springer Nature |
Total Pages | : 421 |
Release | : 2019-12-03 |
Genre | : Mathematics |
ISBN | : 3030320111 |
The 2017 PIMS-CRM Summer School in Probability was held at the Pacific Institute for the Mathematical Sciences (PIMS) at the University of British Columbia in Vancouver, Canada, during June 5-30, 2017. It had 125 participants from 20 different countries, and featured two main courses, three mini-courses, and twenty-nine lectures. The lecture notes contained in this volume provide introductory accounts of three of the most active and fascinating areas of research in modern probability theory, especially designed for graduate students entering research: Scaling limits of random trees and random graphs (Christina Goldschmidt) Lectures on the Ising and Potts models on the hypercubic lattice (Hugo Duminil-Copin) Extrema of the two-dimensional discrete Gaussian free field (Marek Biskup) Each of these contributions provides a thorough introduction that will be of value to beginners and experts alike.
Field Theory, The Renormalization Group, And Critical Phenomena: Graphs To Computers (3rd Edition)
Author | : Daniel J Amit |
Publisher | : World Scientific Publishing Company |
Total Pages | : 568 |
Release | : 2005-06-21 |
Genre | : Science |
ISBN | : 9813102071 |
This volume links field theory methods and concepts from particle physics with those in critical phenomena and statistical mechanics, the development starting from the latter point of view. Rigor and lengthy proofs are trimmed by using the phenomenological framework of graphs, power counting, etc., and field theoretic methods with emphasis on renormalization group techniques. Non-perturbative methods and numerical simulations are introduced in this new edition. Abundant references to research literature complement this matter-of-fact approach. The book introduces quantum field theory to those already grounded in the concepts of statistical mechanics and advanced quantum theory, with sufficient exercises in each chapter for use as a textbook in a one-semester graduate course.The following new chapters are included:I. Real Space MethodsII. Finite Size ScalingIII. Monte Carlo Methods. Numerical Field Theory
Progress in High-Dimensional Percolation and Random Graphs
Author | : Markus Heydenreich |
Publisher | : Springer |
Total Pages | : 285 |
Release | : 2017-11-22 |
Genre | : Mathematics |
ISBN | : 3319624733 |
This text presents an engaging exposition of the active field of high-dimensional percolation that will likely provide an impetus for future work. With over 90 exercises designed to enhance the reader’s understanding of the material, as well as many open problems, the book is aimed at graduate students and researchers who wish to enter the world of this rich topic. The text may also be useful in advanced courses and seminars, as well as for reference and individual study. Part I, consisting of 3 chapters, presents a general introduction to percolation, stating the main results, defining the central objects, and proving its main properties. No prior knowledge of percolation is assumed. Part II, consisting of Chapters 4–9, discusses mean-field critical behavior by describing the two main techniques used, namely, differential inequalities and the lace expansion. In Parts I and II, all results are proved, making this the first self-contained text discussing high-dime nsional percolation. Part III, consisting of Chapters 10–13, describes recent progress in high-dimensional percolation. Partial proofs and substantial overviews of how the proofs are obtained are given. In many of these results, the lace expansion and differential inequalities or their discrete analogues are central. Part IV, consisting of Chapters 14–16, features related models and further open problems, with a focus on the big picture.
Probability and Phase Transition
Author | : G.R. Grimmett |
Publisher | : Springer Science & Business Media |
Total Pages | : 334 |
Release | : 2013-04-17 |
Genre | : Science |
ISBN | : 9401583269 |
This volume describes the current state of knowledge of random spatial processes, particularly those arising in physics. The emphasis is on survey articles which describe areas of current interest to probabilists and physicists working on the probability theory of phase transition. Special attention is given to topics deserving further research. The principal contributions by leading researchers concern the mathematical theory of random walk, interacting particle systems, percolation, Ising and Potts models, spin glasses, cellular automata, quantum spin systems, and metastability. The level of presentation and review is particularly suitable for postgraduate and postdoctoral workers in mathematics and physics, and for advanced specialists in the probability theory of spatial disorder and phase transition.