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Gibbs measures on Cayley trees [[electronic resource] /] / Utkir A. Rozikov
| Gibbs measures on Cayley trees [[electronic resource] /] / Utkir A. Rozikov |
| Autore | Rozikov Utkir A. <1970-> |
| Pubbl/distr/stampa | Singapore ; ; Hackensack, N.J., : World Scientific, 2013 |
| Descrizione fisica | 1 online resource (404 p.) |
| Disciplina | 519.2 |
| Soggetto topico |
Probability measures
Distribution (Probability theory) |
| Soggetto genere / forma | Electronic books. |
| ISBN | 981-4513-38-5 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Preface; 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; 1.3.1 Subgroups of infinite index; 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; 2.1.1 Configuration space; 2.1.2 Hamiltonian; 2.1.3 The ground state; 2.1.4 Gibbs measure; 2.2 A functional equation for the Ising model; 2.2.1 Hamiltonian of the Ising model; 2.2.2 Finite dimensional distributions
2.3 Periodic Gibbs measures of the Ising model2.3.1 Translation-invariant measures of the Ising model; 2.3.1.1 Ferromagnetic case; 2.3.1.2 Anti-ferromagnetic case; 2.3.2 Periodic (non-translation-invariant) measures; 2.4 Weakly periodic Gibbs measures; 2.4.1 The case of index two; 2.4.2 The case of index four; 2.5 Extremality of the disordered Gibbs measure; 2.6 Uncountable sets of non-periodic Gibbs measures; 2.6.1 Bleher-Ganikhodjaev construction; 2.6.2 Zachary construction; 2.7 New Gibbs measures; 2.8 Free energies; 2.9 Ising model with an external field 3. Ising type models with competing interactions3.1 Vannimenus model; 3.1.1 Definitions and equations; 3.1.2 Dynamics of F; 3.1.2.1 Fixed points; 3.1.3 Periodic points; 3.1.4 Exact values; 3.1.5 Remarks; 3.2 A model with four competing interactions; 3.2.1 The model; 3.2.2 The functional equation; 3.2.3 Translation-invariant Gibbs measures: phase transition; 3.2.4 Periodic Gibbs measures; 3.2.5 Non-periodic Gibbs measures; 4. Information ow on trees; 4.1 Definitions and their equivalency; 4.1.1 Equivalent definitions; 4.2 Symmetric binary channels: the Ising model 4.2.1 Reconstruction algorithms4.2.2 Census solvability; 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; 5.2.1 Anti-ferromagnetic case; 5.2.2 Ferromagnetic case; 5.2.2.1 Case: k = 2, q = 3; 5.2.2.2 The general case: k 2, q 2; 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 model6.2.1 The critical value 1cr; 6.2.2 Periodic SGMs; 6.2.3 Non-periodic SGMs; 6.3 Four-state SOS model; 6.3.1 Translation-invariant measures; 6.3.2 Construction of periodic SGMs; 6.3.3 Uncountable set non-periodic SGMs; 7. Models with hard constraints; 7.1 Definitions; 7.1.1 Gibbs measures; 7.2 Two-state hard core model; 7.2.1 Construction of splitting (simple) Gibbs measures; 7.2.2 Uniqueness of a translation-invariant splitting Gibbs measure; 7.2.3 Periodic hard core splitting Gibbs measures; 7.2.4 Extremality of the translation-invariant splitting Gibbs measure 7.2.5 Weakly periodic Gibbs measures |
| Record Nr. | UNINA-9910464104103321 |
Rozikov Utkir A. <1970->
|
||
| Singapore ; ; Hackensack, N.J., : World Scientific, 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Gibbs measures on Cayley trees / / Utkir A. Rozikov, Institute of Mathematics, Uzbekistan
| Gibbs measures on Cayley trees / / Utkir A. Rozikov, Institute of Mathematics, Uzbekistan |
| Autore | Rozikov Utkir A. <1970-> |
| Pubbl/distr/stampa | Singapore ; ; Hackensack, N.J., : World Scientific, 2013 |
| Descrizione fisica | 1 online resource (xviii, 385 pages) : illustrations |
| Disciplina | 519.2 |
| Collana | Gale eBooks |
| Soggetto topico |
Probability measures
Distribution (Probability theory) |
| ISBN | 981-4513-38-5 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Preface; 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; 1.3.1 Subgroups of infinite index; 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; 2.1.1 Configuration space; 2.1.2 Hamiltonian; 2.1.3 The ground state; 2.1.4 Gibbs measure; 2.2 A functional equation for the Ising model; 2.2.1 Hamiltonian of the Ising model; 2.2.2 Finite dimensional distributions
2.3 Periodic Gibbs measures of the Ising model2.3.1 Translation-invariant measures of the Ising model; 2.3.1.1 Ferromagnetic case; 2.3.1.2 Anti-ferromagnetic case; 2.3.2 Periodic (non-translation-invariant) measures; 2.4 Weakly periodic Gibbs measures; 2.4.1 The case of index two; 2.4.2 The case of index four; 2.5 Extremality of the disordered Gibbs measure; 2.6 Uncountable sets of non-periodic Gibbs measures; 2.6.1 Bleher-Ganikhodjaev construction; 2.6.2 Zachary construction; 2.7 New Gibbs measures; 2.8 Free energies; 2.9 Ising model with an external field 3. Ising type models with competing interactions3.1 Vannimenus model; 3.1.1 Definitions and equations; 3.1.2 Dynamics of F; 3.1.2.1 Fixed points; 3.1.3 Periodic points; 3.1.4 Exact values; 3.1.5 Remarks; 3.2 A model with four competing interactions; 3.2.1 The model; 3.2.2 The functional equation; 3.2.3 Translation-invariant Gibbs measures: phase transition; 3.2.4 Periodic Gibbs measures; 3.2.5 Non-periodic Gibbs measures; 4. Information ow on trees; 4.1 Definitions and their equivalency; 4.1.1 Equivalent definitions; 4.2 Symmetric binary channels: the Ising model 4.2.1 Reconstruction algorithms4.2.2 Census solvability; 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; 5.2.1 Anti-ferromagnetic case; 5.2.2 Ferromagnetic case; 5.2.2.1 Case: k = 2, q = 3; 5.2.2.2 The general case: k 2, q 2; 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 model6.2.1 The critical value 1cr; 6.2.2 Periodic SGMs; 6.2.3 Non-periodic SGMs; 6.3 Four-state SOS model; 6.3.1 Translation-invariant measures; 6.3.2 Construction of periodic SGMs; 6.3.3 Uncountable set non-periodic SGMs; 7. Models with hard constraints; 7.1 Definitions; 7.1.1 Gibbs measures; 7.2 Two-state hard core model; 7.2.1 Construction of splitting (simple) Gibbs measures; 7.2.2 Uniqueness of a translation-invariant splitting Gibbs measure; 7.2.3 Periodic hard core splitting Gibbs measures; 7.2.4 Extremality of the translation-invariant splitting Gibbs measure 7.2.5 Weakly periodic Gibbs measures |
| Record Nr. | UNINA-9910787695503321 |
|
Rozikov Utkir A. <1970->
|
||
| Singapore ; ; Hackensack, N.J., : World Scientific, 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Gibbs measures on Cayley trees / / Utkir A. Rozikov, Institute of Mathematics, Uzbekistan
| Gibbs measures on Cayley trees / / Utkir A. Rozikov, Institute of Mathematics, Uzbekistan |
| Autore | Rozikov Utkir A. <1970-> |
| Pubbl/distr/stampa | Singapore ; ; Hackensack, N.J., : World Scientific, 2013 |
| Descrizione fisica | 1 online resource (xviii, 385 pages) : illustrations |
| Disciplina | 519.2 |
| Collana | Gale eBooks |
| Soggetto topico |
Probability measures
Distribution (Probability theory) |
| ISBN | 981-4513-38-5 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Preface; 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; 1.3.1 Subgroups of infinite index; 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; 2.1.1 Configuration space; 2.1.2 Hamiltonian; 2.1.3 The ground state; 2.1.4 Gibbs measure; 2.2 A functional equation for the Ising model; 2.2.1 Hamiltonian of the Ising model; 2.2.2 Finite dimensional distributions
2.3 Periodic Gibbs measures of the Ising model2.3.1 Translation-invariant measures of the Ising model; 2.3.1.1 Ferromagnetic case; 2.3.1.2 Anti-ferromagnetic case; 2.3.2 Periodic (non-translation-invariant) measures; 2.4 Weakly periodic Gibbs measures; 2.4.1 The case of index two; 2.4.2 The case of index four; 2.5 Extremality of the disordered Gibbs measure; 2.6 Uncountable sets of non-periodic Gibbs measures; 2.6.1 Bleher-Ganikhodjaev construction; 2.6.2 Zachary construction; 2.7 New Gibbs measures; 2.8 Free energies; 2.9 Ising model with an external field 3. Ising type models with competing interactions3.1 Vannimenus model; 3.1.1 Definitions and equations; 3.1.2 Dynamics of F; 3.1.2.1 Fixed points; 3.1.3 Periodic points; 3.1.4 Exact values; 3.1.5 Remarks; 3.2 A model with four competing interactions; 3.2.1 The model; 3.2.2 The functional equation; 3.2.3 Translation-invariant Gibbs measures: phase transition; 3.2.4 Periodic Gibbs measures; 3.2.5 Non-periodic Gibbs measures; 4. Information ow on trees; 4.1 Definitions and their equivalency; 4.1.1 Equivalent definitions; 4.2 Symmetric binary channels: the Ising model 4.2.1 Reconstruction algorithms4.2.2 Census solvability; 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; 5.2.1 Anti-ferromagnetic case; 5.2.2 Ferromagnetic case; 5.2.2.1 Case: k = 2, q = 3; 5.2.2.2 The general case: k 2, q 2; 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 model6.2.1 The critical value 1cr; 6.2.2 Periodic SGMs; 6.2.3 Non-periodic SGMs; 6.3 Four-state SOS model; 6.3.1 Translation-invariant measures; 6.3.2 Construction of periodic SGMs; 6.3.3 Uncountable set non-periodic SGMs; 7. Models with hard constraints; 7.1 Definitions; 7.1.1 Gibbs measures; 7.2 Two-state hard core model; 7.2.1 Construction of splitting (simple) Gibbs measures; 7.2.2 Uniqueness of a translation-invariant splitting Gibbs measure; 7.2.3 Periodic hard core splitting Gibbs measures; 7.2.4 Extremality of the translation-invariant splitting Gibbs measure 7.2.5 Weakly periodic Gibbs measures |
| Record Nr. | UNINA-9910812965003321 |
|
Rozikov Utkir A. <1970->
|
||
| Singapore ; ; Hackensack, N.J., : World Scientific, 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||