R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

Elements of the Theory of Functions and Functional Analysis, Volume 1. Me and Normed Spaces.

In this paper we construct several models with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k≥2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.

Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

The purpose of this review paper is to present systematically all known results on Gibbs measures on Cayley trees (Bethe lattices). There are about 150 papers which contain mathematically rigorous results about Gibbs measures on Cayley trees. This review is mainly based on the recently published mathematical papers. 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. We discuss all the mentioned methods which were developed recently. Thus, the paper informs the reader about what is (mathematically) done in the theory of Gibbs measures on trees and about where the corresponding results were published. We only give proofs which were not published in literature. Moreover, we give several open problems.

Gibbs measures on cayley trees: results and open problems

We consider probability measures on a space S(^A) (where S and A are countable and the σ-field is the natural one) which are Markov random fields with respect to a given neighbour relation ~ on A. In particular, we study the set G(II) of Markov random fields corresponding to a given Markov specification II, i.e. to a consistent family of "Markov" conditional probability distributions associated with the finite subsets of A. First, we review the relation between II and G(II). We consider also the representation of II by a family of interaction functions associated with the simplices of the graph (A,~) , together with some related problems. The rest of the thesis is concerned with the case where (A,~) is a tree. We define Markov chains on and consider their relation to the wider class of Markov random fields. We then derive analytical methods for the study of the set M(II) of Markov chains in G(II). These results are applied to homogeneous Markov specifications on regular infinite trees. Finally, we consider Markov specifications which are either attractive or repulsive with respect to a total ordering on S. For these we obtain quite strong results, including an exact condition for G(II) to contain precisely one element. We thereby generalise results obtained by Preston and Spitzer for binary S.

Markov random fields and Markov chains on trees

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order $k\geqslant 1$. It is known that the ‘splitting Gibbs measures’ of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geqslant 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.

Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

In the space L
 2(T
 ν
 ×T
 ν
 ), where T
 ν
 is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrodinger operator Ĥ = H0 + H1 + H2, where H0 is the operator of multiplication by a function and H1 and H2 are partial integral operators. We prove several theorems concerning the essential spectrum of Ĥ. We study the discrete and essential spectra of the Hamiltonians Ht and h arising in the Hubbard model on the three-dimensional lattice.

A discrete “three-particle” Schrödinger operator in the Hubbard model

We study the position of the essential spectrum of a three-body Schrodinger operator H. We evaluate the lower boundary of the essential spectrum of H and prove that the number of eigenvalues located below the lower edge of the essential spectrum in the H model is finite.

Essential and discrete spectra of the three-particle Schrödinger operator on a lattice

In this paper we consider a model with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k \geq 2$. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For $k=2$ and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e. there are phase transitions).

Phase Transitions for a model with uncountable set of spin values on a Cayley tree

In this paper we consider a model with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 2. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For k = 2 and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e., there are phase transitions).

Yu. Kh. Eshkabilov

Papers

Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

A discrete “three-particle” Schrödinger operator in the Hubbard model

Essential and discrete spectra of the three-particle Schrödinger operator on a lattice

Phase Transitions for a model with uncountable set of spin values on a Cayley tree

Phase transitions for a model with uncountable set of spin values on a Cayley tree