The Random Cluster Model On The Tree
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| 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.
| 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.
| Author | : |
| Publisher | : Elsevier |
| Total Pages | : 337 |
| Release | : 2000-09-15 |
| Genre | : Science |
| ISBN | : 0080538754 |
The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. No longer an area of specialist interest, it has acquired a central focus in condensed matter studies. The major aim of this serial is to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments.The two review articles in this volume complement each other in a remarkable way. Both deal with what might be called the modern geometricapproach to the properties of macroscopic systems. The first article by Georgii (et al.) describes how recent advances in the application ofgeometric ideas leads to a better understanding of pure phases and phase transitions in equilibrium systems. The second article by Alava (et al.)deals with geometrical aspects of multi-body systems in a hands-on way, going beyond abstract theory to obtain practical answers. Thecombination of computers and geometrical ideas described in this volume will doubtless play a major role in the development of statisticalmechanics in the twenty-first century.
| Author | : Russell Lyons |
| Publisher | : Cambridge University Press |
| Total Pages | : 1023 |
| Release | : 2017-01-20 |
| Genre | : Mathematics |
| ISBN | : 1316785335 |
Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.
| Author | : Rick Durrett |
| Publisher | : Cambridge University Press |
| Total Pages | : 203 |
| Release | : 2010-05-31 |
| Genre | : Mathematics |
| ISBN | : 1139460889 |
The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph in which each site is connected to k close neighbors, but also has long-range connections. At a similar time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers have led to an explosion of research. The purpose of this book is to use a wide variety of mathematical argument to obtain insights into the properties of these graphs. A unique feature is the interest in the dynamics of process taking place on the graph in addition to their geometric properties, such as connectedness and diameter.
| Author | : James E. Corter |
| Publisher | : SAGE |
| Total Pages | : 76 |
| Release | : 1996-04-02 |
| Genre | : Medical |
| ISBN | : 9780803957077 |
This book describes how matrices of similarities or associations among entities can be modelled using trees in order to explain some of the issues that arise in performing similarity relations analyses and interpreting the results correctly.
| Author | : David Banks |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 642 |
| Release | : 2011-01-07 |
| Genre | : Language Arts & Disciplines |
| ISBN | : 3642171036 |
This volume describes new methods with special emphasis on classification and cluster analysis. These methods are applied to problems in information retrieval, phylogeny, medical diagnosis, microarrays, and other active research areas.
| Author | : Utkir A Rozikov |
| Publisher | : World Scientific |
| Total Pages | : 367 |
| Release | : 2022-07-28 |
| Genre | : Mathematics |
| ISBN | : 9811251258 |
This book presents recently obtained mathematical results on Gibbs measures of the q-state Potts model on the integer lattice and on Cayley trees. It also illustrates many applications of the Potts model to real-world situations in biology, physics, financial engineering, medicine, and sociology, as well as in some examples of alloy behavior, cell sorting, flocking birds, flowing foams, and image segmentation.Gibbs measure is one of the important measures in various problems of probability theory and statistical mechanics. It is a measure associated with the Hamiltonian of a biological or physical system. Each Gibbs measure gives a state of the system.The main problem for a given Hamiltonian on a countable lattice is to describe all of its possible Gibbs measures. The existence of some values of parameters at which the uniqueness of Gibbs measure switches to non-uniqueness is interpreted as a phase transition.This book informs the reader about what has been (mathematically) done in the theory of Gibbs measures of the Potts model and the numerous applications of the Potts model. The main aim is to facilitate the readers (in mathematical biology, statistical physics, applied mathematics, probability and measure theory) to progress into an in-depth understanding by giving a systematic review of the theory of Gibbs measures of the Potts model and its applications.
| Author | : Fa Yueh Wu |
| Publisher | : World Scientific |
| Total Pages | : 661 |
| Release | : 2009-03-03 |
| Genre | : Science |
| ISBN | : 9814471224 |
This unique volume provides a comprehensive overview of exactly solved models in statistical mechanics by looking at the scientific achievements of F Y Wu in this and related fields, which span four decades of his career. The book is organized into topics ranging from lattice models in condensed matter physics to graph theory in mathematics, and includes the author's pioneering contributions. Through insightful commentaries, the author presents an overview of each of the topics and an insider's look at how crucial developments emerged. With the inclusion of important pedagogical review articles by the author, Exactly Solved Models is an indispensable learning tool for graduate students, and an essential reference and source book for researchers in physics and mathematics as well as historians of science.
| Author | : Geoffrey R. Grimmett |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 459 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 3662039818 |
Percolation theory is the study of an idealized random medium in two or more dimensions. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. Much new material appears in this second edition including dynamic and static renormalization, strict inequalities between critical points, a sketch of the lace expansion, and several essays on related fields and applications.
