An Introduction To Mathematical Billiards

Author : Utkir A Rozikov
Publisher : World Scientific
ISBN 13 : 9814513393
Total Pages : 404 pages
Book Rating : 4.8/5 (145 download)

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Download or read book Gibbs Measures on Cayley Trees written by Utkir A Rozikov and published by World Scientific. This book was released on 2013-07-11 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to present systematically all known mathematical results on Gibbs measures on Cayley trees (Bethe lattices). The Gibbs measure is a probability measure, which has been an important object in many problems of probability theory and statistical mechanics. It is the measure associated with the Hamiltonian of a physical system (a model) and generalizes the notion of a canonical ensemble. More importantly, when the Hamiltonian can be written as a sum of parts, the Gibbs measure has the Markov property (a certain kind of statistical independence), thus leading to its widespread appearance in many problems outside of physics such as biology, Hopfield networks, Markov networks, and Markov logic networks. Moreover, the Gibbs measure is the unique measure that maximizes the entropy for a given expected energy. The method used for the description of Gibbs measures on Cayley trees is the method of Markov random field theory and recurrent equations of this theory, but the modern theory of Gibbs measures on trees uses new tools such as group theory, information flows on trees, node-weighted random walks, contour methods on trees, and nonlinear analysis. This book discusses all the mentioned methods, which were developed recently. Contents:Group Representation of the Cayley TreeIsing Model on the Cayley TreeIsing Type Models with Competing InteractionsInformation Flow on TreesThe Potts ModelThe Solid-on-Solid ModelModels with Hard ConstraintsPotts Model with Countable Set of Spin ValuesModels with Uncountable Set of Spin ValuesContour Arguments on Cayley TreesOther Models Readership: Researchers in mathematical physics, statistical physics, probability and measure theory. Keywords:Cayley Tree;Configuration;Hamiltonian;Temperature;Gibbs MeasureKey Features:The book is for graduate, post-graduate students and researchers. This is the first book concerning Gibbs measures on Cayley treesIt can be used to teach special courses like “Gibbs measures on countable graphs”, “Models of statistical physics”, “Phase transitions and thermodynamics” and many related coursesReviews: “The extensive commentaries and references which follow are as valuable as the mathematical text. At the end of each chapter, the author gives extensive commentaries and a list of references to the literature, including very recent ones. The reader may find useful and insightful open problems concluding the end of each chapter. The book is written from the mathematician's point of view and its addressees are professionals in statistical mechanics and mathematical physics.” Zentralblatt MATH