| Description |
1 online resource |
| 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 non-periodic 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. q-ary symmetric channels: the Potts model -- 5. The Potts model. 5.1. The Hamiltonian and vector-valued functional equation. 5.2. Translation-invariant 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 Solid-on-Solid model. 6.1. The model and a system of vector-valued functional equations. 6.2. Three-state SOS model. 6.3. Four-state SOS model -- 7. Models with hard constraints. 7.1. Definitions. 7.2. Two-state hard core model. 7.3. Node-weighted random walk as a tool. 7.4. A Gibbs measure associated to a k-branching nodeweighted random walk. 7.5. Cases of uniqueness of Gibbs measure. 7.6. Non-uniqueness of Gibbs measure: sterile and fertile graphs. 7.7. Fertile three-state hard core models. 7.8. Eight state hard-core 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. Translation-invariant solutions. 8.3. Exponential solutions -- 9. Models with uncountable set of spin values. 9.1. Definitions. 9.2. An integral equation. 9.3. Translational-invariant solutions. 9.4. A sufficient condition of uniqueness. 9.5. Examples of Hamiltonians with non-unique Gibbs measure -- 10. Contour arguments on Cayley trees. 10.1. One-dimensional models. 10.2. q-component models. 10.3. An Ising model with competing two-step interactions. 10.4. Finite-range models: general contours -- 11. Other models. 11.1. Inhomogeneous Ising model. 11.2. Random field Ising model. 11.3. Ashkin-Teller 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, node-weighted random walks, contour methods on trees, and nonlinear analysis. This book discusses all the mentioned methods, which were developed recently |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Probability measures.
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Distribution (Probability theory)
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distribution (statistics-related concept)
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MATHEMATICS -- Probability & Statistics -- General.
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Distribution (Probability theory)
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Probability measures
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| Form |
Electronic book
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| ISBN |
9789814513388 |
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9814513385 |
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