Gibbs measures on Cayley trees:
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| Format: | Elektronisch E-Book |
| Sprache: | English |
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[Hackensack] New Jersey
World Scientific
2013
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| Online-Zugang: | FAW01 FAW02 Volltext |
| Beschreibung: | Includes bibliographical references and index 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 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 |
| Beschreibung: | 1 Online-Ressource (pages cm.) |
| ISBN: | 9789814513371 9789814513388 9814513377 9814513385 |
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| 245 | 1 | 0 | |a Gibbs measures on Cayley trees |c by Utkir A. Rozikov (Institute of Mathematics, Uzbekistan) |
| 264 | 1 | |a [Hackensack] New Jersey |b World Scientific |c 2013 | |
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| 500 | |a Includes bibliographical references and index | ||
| 500 | |a 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 -- | ||
| 500 | |a - 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 -- | ||
| 500 | |a - 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 | ||
| 500 | |a 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 | ||
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| 650 | 7 | |a Distribution (Probability theory) |2 fast | |
| 650 | 7 | |a Probability measures |2 fast | |
| 650 | 4 | |a Probability measures | |
| 650 | 4 | |a Distribution (Probability theory) | |
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Datensatz im Suchindex
| _version_ | 1804175438147223552 |
|---|---|
| any_adam_object | |
| author | Rozikov, Utkir A. |
| author_facet | Rozikov, Utkir A. |
| author_role | aut |
| author_sort | Rozikov, Utkir A. |
| author_variant | u a r ua uar |
| building | Verbundindex |
| bvnumber | BV043062277 |
| collection | ZDB-4-EBA |
| ctrlnum | (OCoLC)855022908 (DE-599)BVBBV043062277 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | DE-604.BV043062277 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:16:17Z |
| institution | BVB |
| isbn | 9789814513371 9789814513388 9814513377 9814513385 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028486469 |
| oclc_num | 855022908 |
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| owner | DE-1046 DE-1047 |
| owner_facet | DE-1046 DE-1047 |
| physical | 1 Online-Ressource (pages cm.) |
| psigel | ZDB-4-EBA ZDB-4-EBA FAW_PDA_EBA |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Rozikov, Utkir A. Verfasser aut Gibbs measures on Cayley trees by Utkir A. Rozikov (Institute of Mathematics, Uzbekistan) [Hackensack] New Jersey World Scientific 2013 1 Online-Ressource (pages cm.) txt rdacontent c rdamedia cr rdacarrier Includes bibliographical references and index 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 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 MATHEMATICS / Probability & Statistics / General bisacsh Distribution (Probability theory) fast Probability measures fast Probability measures Distribution (Probability theory) Cayley-Baum (DE-588)4466547-7 gnd rswk-swf Gibbs-Maß (DE-588)4157328-6 gnd rswk-swf Cayley-Baum (DE-588)4466547-7 s Gibbs-Maß (DE-588)4157328-6 s 1\p DE-604 http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=622028 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Rozikov, Utkir A. Gibbs measures on Cayley trees MATHEMATICS / Probability & Statistics / General bisacsh Distribution (Probability theory) fast Probability measures fast Probability measures Distribution (Probability theory) Cayley-Baum (DE-588)4466547-7 gnd Gibbs-Maß (DE-588)4157328-6 gnd |
| subject_GND | (DE-588)4466547-7 (DE-588)4157328-6 |
| title | Gibbs measures on Cayley trees |
| title_auth | Gibbs measures on Cayley trees |
| title_exact_search | Gibbs measures on Cayley trees |
| title_full | Gibbs measures on Cayley trees by Utkir A. Rozikov (Institute of Mathematics, Uzbekistan) |
| title_fullStr | Gibbs measures on Cayley trees by Utkir A. Rozikov (Institute of Mathematics, Uzbekistan) |
| title_full_unstemmed | Gibbs measures on Cayley trees by Utkir A. Rozikov (Institute of Mathematics, Uzbekistan) |
| title_short | Gibbs measures on Cayley trees |
| title_sort | gibbs measures on cayley trees |
| topic | MATHEMATICS / Probability & Statistics / General bisacsh Distribution (Probability theory) fast Probability measures fast Probability measures Distribution (Probability theory) Cayley-Baum (DE-588)4466547-7 gnd Gibbs-Maß (DE-588)4157328-6 gnd |
| topic_facet | MATHEMATICS / Probability & Statistics / General Distribution (Probability theory) Probability measures Cayley-Baum Gibbs-Maß |
| url | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=622028 |
| work_keys_str_mv | AT rozikovutkira gibbsmeasuresoncayleytrees |