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001    855022908 
003    OCoLC 
005    20180309083628.8 
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007    cr cnu---unuuu 
008    130805s2013    nju     ob    001 0 eng d 
019    855505006|a861257286 
020    9789814513388|q(electronic bk.) 
020    9814513385|q(electronic bk.) 
020    |z9789814513371 
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049    MTUM 
050  4 QA273.6|b.R69 2013eb 
072  7 MAT|x029000|2bisacsh 
082 04 519.2|223 
100 1  Rozikov, Utkir A.,|d1970-|0http://id.loc.gov/authorities/
       names/n2013028504|eauthor. 
245 10 Gibbs measures on Cayley trees /|cby Utkir A. Rozikov 
       (Institute of Mathematics, Uzbekistan). 
264  1 [Hackensack] New Jersey :|bWorld Scientific,|c2013. 
300    1 online resource 
336    text|btxt|2rdacontent 
337    computer|bc|2rdamedia 
338    online resource|bcr|2rdacarrier 
340    |gpolychrome|2rdacc|0http://rdaregistry.info/termList/
       RDAColourContent/1003 
347    text file|2rdaft|0http://rdaregistry.info/termList/
       fileType/1002 
504    Includes bibliographical references and index. 
505 0  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. 
520    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. 
588 0  Print version record. 
650  0 Probability measures.|0http://id.loc.gov/authorities/
       subjects/sh92001869 
650  0 Distribution (Probability theory)|0http://id.loc.gov/
       authorities/subjects/sh85038545 
655  4 Electronic books. 
776 08 |iPrint version:|aRozikov, Utkir A., 1970-|tGibbs measures
       on Cayley trees.|d[Hackensack] New Jersey : World 
       Scientific, 2013|z9789814513371|w(DLC)  2013014066
       |w(OCoLC)844073549 
956 40 |uhttp://proxy.opal-libraries.org/login?url=http://
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       db=nlebk&AN=622028|zView online 
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