Contents -- Preface -- 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 model -- 2.3.1 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 interactions -- 3.1 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 algorithms.

4.2.2 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 model -- 6.2.1 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 -- 7.2.6 The model with two fugacities -- 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 -- 7.6.1 The Asymmetric Graphs -- 7.6.2 The Wand and the Hinge -- 7.6.3 The Stick -- 7.6.4 The hinge -- 7.7 Fertile three-state hard core models -- 7.7.1 System of functional equations -- 7.7.2 Translation-invariant Gibbs measures -- 7.7.2.1 The Case wrench -- 7.7.2.2 The case hinge -- 7.7.2.3 The Case wand -- 7.7.2.4 The case pipe.

7.7.3 Periodic Gibbs measures -- 7.7.3.1 The case wrench -- 7.7.3.2 The case hinge -- 7.7.4 Non-Periodic Gibbs measures: the case hinge -- 7.8 Eight state hard-core model associated to a model with interaction radius two -- 7.8.1 The system of functional equations -- 7.8.2 Translation-invariant solutions -- 7.8.3 Periodic solutions -- 8. Potts model with countable set of spin values -- 8.1 An infinite system of functional equations -- 8.2 Translation-invariant solutions -- 8.2.1 The set of solutions {ui} with j=1 uj = -- 8.2.2 The set of solutions with j=1 uj 1 -- 8.2.2.2 Case 1 -- 8.3 Exponential solutions -- 8.3.1 Case > 1 -- 8.3.2 Case 1 -- 9. Models with uncountable set of spin values -- 9.1 Definitions -- 9.2 An integral equation -- 9.2.1 The Potts model with uncountable spin values -- 9.3 Translational-invariant solutions -- 9.3.1 Case k = 1 -- 9.3.2 Case k 2 -- 9.4 A sufficient condition of uniqueness -- 9.4.1 The Hammerstein's non-linear equation -- 9.4.2 The uniqueness of fixed point of the operators Ak and Hk -- 9.4.3 Physical interpretation -- 9.5 Examples of Hamiltonians with non-unique Gibbs measure -- 9.5.1 Case k = 2 -- 9.5.2 Case k = 3 -- 9.5.3 Case k 4 -- 10. Contour arguments on Cayley trees -- 10.1 One-dimensional models -- 10.1.1 Phase transition -- 10.1.2 Partition functions -- 10.1.2.1 Partition function of "+" and " "-boundary conditions -- 10.1.2.2 Crystal partition functions -- 10.1.3 Phase-separation point -- 10.2 q-component models -- 10.2.1 Contours for the q-component models on the Cayley tree -- 10.2.2 Additional properties of the contours -- 10.2.3 The contour Hamiltonian -- 10.2.4 The Potts model -- 10.2.5 The SOS model -- 10.3 An Ising model with competing two-step interactions -- 10.3.1 Ground states -- 10.3.2 Weakly periodic ground states -- 10.3.2.1 Case of index two.

10.3.2.2 Case of index four -- 10.3.3 The Peierls condition -- 10.3.4 Contours and Gibbs measures -- 10.4 Finite-range models: general contours -- 10.4.1 Configuration space and the model -- 10.4.2 The assumptions and Peierls condition -- 10.4.3 Contours -- 10.4.4 Non-uniqueness of Gibbs measure -- 10.4.5 Examples -- 10.4.5.1 q-component models -- 10.4.5.2 The Potts model with competing interactions -- 10.4.5.3 A model with the interaction radius r 1 -- 11. Other models -- 11.1 Inhomogeneous Ising model -- 11.2 Random field Ising model -- 11.3 Ashkin-Teller model -- 11.3.1 Paramagnetic fixed point -- 11.3.2 Non-trivial fixed points -- 11.4 Spin glass model -- 11.5 Abelian sandpile model -- 11.6 Z(M) (or clock) models -- 11.6.1 The model and equations -- 11.6.2 Phases of Z(M) 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 -- 11.10.1 Real values -- 11.10.2 Quantum case -- 11.10.3 p-adic values -- Bibliography -- Index.