Gibbs measures on Cayley trees (Q2849642)

The Ising and Potts models are related to a number of outstanding problems in statistical and mathematical physics, and in graph theory. It is well known that the Cayley tree is not a realistic lattice; however, its amazing topology makes the exact calculation of various quantities possible. For many problems, the solution on a tree is much simpler than on a regular lattice and is equivalent to the standard Bethe-Peierls theory. Therefore, the results presented in the book can lead new more realistic approaches. It is well known that one of the main problems of statistical physics is to describe all Gibbs measures corresponding to a given Hamiltonian. The book contains mathematically rigorous results about Gibbs measures of given models on Cayley trees. It studies systematically known mathematical results on Gibbs measures of the Ising or Potts models on Cayley trees. In the book, the author presents a number of theorems related to the description of the general structure of Gibbs measures of the Ising or Potts model on Cayley trees.NEWLINENEWLINEThere are many approaches to derive equations or system equations describing Gibbs measures for lattice models on Cayley trees. One of these approaches is based on recursive equations for partition functions. Another approach is Markov random fields. Naturally both approaches lead to the same equation. In the book, the author deals with both the first and the second approach. Recently, many researchers have used new tools such as group theory, information flow and contour methods on trees to investigate the modern theory of Gibbs measures. In the book, the author informs the reader about what has been mathematically done in the theory of Gibbs measures on trees and where the corresponding results were published.NEWLINENEWLINEThe extensive commentaries and references which follow are as valuable as the mathematical text. At the end of each chapter, the author gives extensive commentaries and a list of references to the literature, including very recent ones. The reader may find useful and insightful open problems concluding the end of each chapter. The book is written from the mathematician's point of view and its addressees are professionals in statistical mechanics and mathematical physics.NEWLINENEWLINEThe book consists of eleven chapters. In Chapter 1, properties of a group representation of the Cayley tree are given and several subgroups of the group are constructed. This chapter contains 5 sections. The structure of a semi-infinite Cayley tree is given. The author studies the partition structures of the Cayley tree with respect to normal subgroups of a free product of \(k+1\) cyclic groups. The results of Chapter 1 are applied in the next chapters to obtain periodic and weak periodic Gibbs measures of statistical mechanics on a Cayley tree.NEWLINENEWLINEIn Chapter 2, general definitions of configuration spaces and Hamiltonian and Gibbs measures are given. The author explains all known results about Gibbs measures of the Ising model on Cayley trees. He gives a complete description of periodic Gibbs measures with respect to any normal subgroup of finite index for the Ising model. The notion of weakly periodic Gibbs measures is introduced. The author shows that the Ising model has at least seven such measures under some conditions on parameters. To obtain extremality of the disordered Gibbs measure, the author gives a proof. By using the Bleher-Ganikhodjaev construction, the author obtains some constructions of non-periodic extreme Gibbs measures for the Ising model on a Cayley tree of arbitrary order. Also, by the Zachary construction, new Gibbs measures of the Ising model on the Cayley tree of order \(k\geq 3\) are described. Lastly, free energies of the Ising model corresponding to all known Gibbs measures are computed and some open problems are given.NEWLINENEWLINEIn Chapter 3, the author studies Vannimenus's model and the Ising model with four competing interactions on a Cayley tree of order two. The non-uniqueness of the Gibbs measure for some parameters of the models is analytically proved. Paramagnetic and ferromagnetic phases are obtained. For the model with four competing interactions, the Gibbs measures are described and the phase transition is studied. Also, a complete description of periodic Gibbs measures for the models and a construction of uncountably many non-periodic extreme Gibbs measures are given. Note that the first section is based on the paper by the author et al. [``Exact solution of a generalized ANNNI model on a Cayley tree'', Preprint, \url{arXiv:1008.3307}].NEWLINENEWLINEIn Chapter 4 (Information flow on trees), a process on a tree \(T\) in which information is transmitted from the root of the tree to all the nodes of the tree is considered. Because their applicability finding regions of extremality of disordered phases for many models in statistical mechanics is important, results and challenges related to the problem are given.NEWLINENEWLINEIn Chapter 5 (\(q\)-state Potts models), the author studies results related to Gibbs measures of the \(q\)-state Potts model on Cayley trees, where the spin takes values in the set \(\Phi :=\{1,2,\ldots,q\}\). The description of such measures is reduced to the solution of a vector-valued functional equation. As mentioned above, one of the approaches to obtain Gibbs measures of any model is based on recursive equations for partition functions. Using this approach, the author shows the existence of an uncountable set of non-transition Gibbs measures of the Potts model on the Cayley tree. It is proved that the Gibbs measures \(\mu^{i}\), \(i=1,2,3\), are extreme. A complete description of periodic and weak periodic Gibbs measures for the \(q\)-state Potts models remains an open problem.NEWLINENEWLINEIn Chapter 6 (The solid-on-solid model), a nearest-neighbor solid-on-solid (SOS) model, with several spin values in the set \(\Phi :=\{1,2,\ldots,m\}\), \(m \geq 2\), and zero external field, on a Cayley tree of order \(k\), is considered. For \(m=2\), in the anti-ferromagnetic case, the author shows that the translation invariant Gibbs measure is unique for all temperatures. For m = 2, in the ferromagnetic case, the number of such measures varies with the temperature: an example of a phase transition is given. As a second result, the author obtains a complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree.NEWLINENEWLINEChapter 7 (Models with hard constraints) is devoted to the examination of Gibbs measures of the models with hard constraints on trees, which is based on the paper [\textit{G. R. Brightwell} and \textit{P. Winkler}, J. Comb. Theory, Ser. B 77, No. 2, 221--262 (1999; Zbl 1026.05028)]. The periodic, non-periodic and weakly periodic Gibbs measures corresponding to the models are studied.NEWLINENEWLINEIn Chapter 8 (Potts model with countable set of spin values), the author considers a nearest-neighbor Potts model with countable spin values in the set of all non-negative integer numbers \(\Phi :=\{1,2,\ldots\}\), and non-zero external field, on a Cayley tree of order \(k\). By computing the solution of some infinite system of equations, the author includes the results based on [\textit{N. N. Ganikhodjaev} and \textit{U. A. Rozikov}, Lett. Math. Phys. 75, No. 2, 99--109 (2006; Zbl 1101.82003)] and [\textit{N. N. Ganikhodjaev}, J. Math. Anal. Appl. 336, No. 1, 693--703 (2007; Zbl 1151.82006)].NEWLINENEWLINEIn Chapter 9 (Models with uncountable set of spin values), the author presents results based on his papers with \textit{Yu. Kh. Eshkabilov} and \textit{F. H. Haydarov} [Math. Phys. Anal. Geom. 16, No. 1, 1--17 (2013; Zbl 1277.82008); J. Stat. Phys. 147, No. 4, 779--794 (2012; Zbl 1252.82026)] and with \textit{Yu. Kh. Eshkobilov} [Math. Phys. Anal. Geom. 13, No. 3, 275--286 (2010; Zbl 1252.82034)], for models with nearest-neighbor interactions and with the set \([0, 1]\) of spin values, on a Cayley tree of order \(k \geq 1\).NEWLINENEWLINEIn Chapter 10 (Contour arguments on Cayley trees), the author presents recently developed contour methods on Cayley trees. Under some conditions on parameters \(I_n\), it is shown that the phase transition occurs for a one-dimensional model with nearest-neighbor interactions. Using a contour argument the author proves the existence of several different Gibbs measures. In this chapter, he includes the results based on his paper with \textit{G. I. Botirov} [Theor. Math. Phys. 153, No. 1, 1423-1433 (2007); translation from Teor. Mat. Fiz. 153, No. 1, 86-97 (2007; Zbl 1157.82308)] and some other papers given in the references of the book.NEWLINENEWLINEIn the final chapter, the author studies several models, such as inhomogeneous Ising, random field Ising, Ashkin-Teller, quantum models and so on, not discussed in previous chapters. Also, a brief description of the differences between classical (real) models and \(p\)-adic models on Cayley trees is given. For example, in his paper with \textit{D. Gandolfo} and \textit{J. Ruiz} [Markov Process. Relat. Fields 18, No. 4, 701-720 (2012; Zbl 1281.82006)] it is shown that a \(p\)-adic hard core model is completely different from a real hard core model. In the \(p\)-adic case there are many unsolved problems, therefore brief comments in the last chapter of the book will inspire researchers in the statistical physics.