The problem of “small-field” phase transitions for Z(M) models on Cayley trees is solved in detail. Phase diagrams for zero field are obtained for M = 2, 3, 4, 5 and 6. As special cases, Potts models are also considered and all phases (not only those in zero field) are identified. The M →...

A general theory for multicomponent spin models with transitive symmetry groups is developed. These symmetry groups are shown to belong to a certain class of “permissible” groups. A number of theorems concerning these groups is proved and large classes of them are explicitly constructed. The...

A number of new results concerning the multicomponent spin systems studied in the first two papers of this series is derived. These results allow for the determination of all such models with number of components M ⩽ 10. There are shown to be 41 such models, each of which is given in terms of...

The Kirkwood-Salsburg and the Mayer-Montroll equations for an arbitrary stable interaction are derived for the case of an exponentially integrable external potential (of which a finite volume is a special case). It is shown that the Mayer-Montroll equation has at least one solution (the...

A study of an Ising model on a variety of Cayley tree-like lattices with its boundary in a magnetic field is made. It is proved rigorously that such models exhibit phase transitions only in the limit of zero field. The spontaneous magnetization far from the boundary is shown to behave as (Tinc...

Nonlinear Kirkwood-Salsburg equations which are parametrized by the density ϱ are derived from the linear ones by elimination of the activity z. Upper bounds on ϱ are derived below which the solution of these equations is unique. Narrow upper and lower bounds on z(ϱ) are obtained as well as...

Some of the multicomponent spin models with transitive symmetry groups derived in the first paper of this series are solved in detail on the Cayley tree pseudo-lattice. The symmetry groups treated are: (i) the group of the cube, L(2)⊗L(4), (ii) the Klein group K(4), equivalent to the...

A mathematically precise definition of the “infinite-volume” Kirkwood-Salsburg operator as a bounded linear operator in a Banach space is given. It is shown that this operator has a bounded inverse for a bounded, stable and regular pair potential. These facts are exploited to establish the...

A generalized Kirkwood-Salsburg equation for a classical system of particles with an arbitrary stable interaction in an external potential σ(x) such that exp (- βσ) is integrable is derived. It is shown that the spectrum of the generalized Kirkwood-Salsburg operator consists of the inverses...

It is shown, that the configurational partition function for a classical system of molecules interacting with nonspherical pair potential is proportionals to the configurational partition function for a system of particles interacting with temperature-dependent spherical k-body potentials (k...