LEADER 05346nam 2200649 a 450 001 9910464104103321 005 20200520144314.0 010 $a981-4513-38-5 035 $a(CKB)2670000000404109 035 $a(EBL)1336552 035 $a(OCoLC)855505006 035 $a(SSID)ssj0000950142 035 $a(PQKBManifestationID)12421508 035 $a(PQKBTitleCode)TC0000950142 035 $a(PQKBWorkID)11005380 035 $a(PQKB)11381837 035 $a(MiAaPQ)EBC1336552 035 $a(WSP)00008841 035 $a(PPN)189428473 035 $a(Au-PeEL)EBL1336552 035 $a(CaPaEBR)ebr10742821 035 $a(CaONFJC)MIL508340 035 $a(EXLCZ)992670000000404109 100 $a20130520d2013 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aGibbs measures on Cayley trees$b[electronic resource] /$fUtkir A. Rozikov 210 $aSingapore ;$aHackensack, N.J. $cWorld Scientific$d2013 215 $a1 online resource (404 p.) 300 $aDescription based upon print version of record. 311 $a1-299-77089-4 311 $a981-4513-37-7 320 $aIncludes bibliographical references and index. 327 $aPreface; 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 327 $a2.3 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 327 $a3. 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 327 $a4.2.1 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 327 $a6.2 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 327 $a7.2.5 Weakly periodic Gibbs measures 330 $aThe 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. Mor 606 $aProbability measures 606 $aDistribution (Probability theory) 608 $aElectronic books. 615 0$aProbability measures. 615 0$aDistribution (Probability theory) 676 $a519.2 700 $aRozikov$b Utkir A.$f1970-$0993395 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910464104103321 996 $aGibbs measures on Cayley trees$92274602 997 $aUNINA