Author 
Rozikov, Utkir A., 1970

Subject 
Probability measures.


Distribution (Probability theory)

Description 
1 online resource 

polychrome rdacc 
Bibliography Note 
Includes bibliographical references and index. 
Contents 
1. Group representation of the Cayley tree. 1.1. Cayley tree. 1.2. A group representation of the Cayley tree. 1.3. Normal subgroups of finite index for the group representation of the Cayley tree. 1.4. Partition structures of the Cayley tree. 1.5. Density of edges in a ball  2. Ising model on the Cayley tree. 2.1. Gibbs measure. 2.2. A functional equation for the Ising model. 2.3. Periodic Gibbs measures of the Ising model. 2.4. Weakly periodic Gibbs measures. 2.5. Extremality of the disordered Gibbs measure. 2.6. Uncountable sets of nonperiodic Gibbs measures. 2.7. New Gibbs measures. 2.8. Free energies. 2.9. Ising model with an external field  3. Ising type models with competing interactions. 3.1. Vannimenus model. 3.2. A model with four competing interactions  4. Information flow on trees. 4.1. Definitions and their equivalency. 4.2. Symmetric binary channels: the Ising model. 4.3. qary symmetric channels: the Potts model  5. The Potts model. 5.1. The Hamiltonian and vectorvalued functional equation. 5.2. Translationinvariant Gibbs measures. 5.3. Extremality of the disordered Gibbs measure: the reconstruction solvability. 5.4. A construction of an uncountable set of Gibbs measures  6. The SolidonSolid model. 6.1. The model and a system of vectorvalued functional equations. 6.2. Threestate SOS model. 6.3. Fourstate SOS model  7. Models with hard constraints. 7.1. Definitions. 7.2. Twostate hard core model. 7.3. Nodeweighted random walk as a tool. 7.4. A Gibbs measure associated to a kbranching nodeweighted random walk. 7.5. Cases of uniqueness of Gibbs measure. 7.6. Nonuniqueness of Gibbs measure: sterile and fertile graphs. 7.7. Fertile threestate hard core models. 7.8. Eight state hardcore model associated to a model with interaction radius two  8. Potts model with countable set of spin values. 8.1. An infinite system of functional equations. 8.2. Translationinvariant solutions. 8.3. Exponential solutions  9. Models with uncountable set of spin values. 9.1. Definitions. 9.2. An integral equation. 9.3. Translationalinvariant solutions. 9.4. A sufficient condition of uniqueness. 9.5. Examples of Hamiltonians with nonunique Gibbs measure  10. Contour arguments on Cayley trees. 10.1. Onedimensional models. 10.2. qcomponent models. 10.3. An Ising model with competing twostep interactions. 10.4. Finiterange models: general contours  11. Other models. 11.1. Inhomogeneous Ising model. 11.2. Random field Ising model. 11.3. AshkinTeller model. 11.4. Spin glass model. 11.5. Abelian sandpile model. 11.6. Z(M) (or clock) models. 11.7. The planar rotator model. 11.8. O(n, 1)model. 11.9. Supersymmetric O(n, 1) model. 11.10. The review of remaining models. 
Summary 
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, nodeweighted random walks, contour methods on trees, and nonlinear analysis. This book discusses all the mentioned methods, which were developed recently. 
Note 
Print version record. 
ISBN 
9789814513388 (electronic bk.) 

9814513385 (electronic bk.) 

9789814513371 

9814513377 
OCLC # 
855022908 
Additional Format 
Print version: Rozikov, Utkir A., 1970 Gibbs measures on Cayley trees. [Hackensack] New Jersey : World Scientific, 2013 9789814513371 (DLC) 2013014066 (OCoLC)844073549 
