The Ising system with an interaction of finite range on the Cayley tree

It is shown that the thermodynamic properties and the distribution functions of the Ising systems on the Cayley tree are generally obtained in terms of the solution of a recurrence formula, when the interaction is of finite range. For the Bethe lattice where the boundary effects are ignored, the properties are given in terms of the solution of a nonlinear eigenvalue problem. It is further shown that a certain approximation in the cluster variation method is exact for this system and the equations determining the parameters occurring in this method are equivalent to the obtained nonlinear eigenvalue problem.