Ising spin systems on Cayley tree-like lattices: Spontaneous magnetization and correlation functions far from the boundary

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 − T)12 near Tc, and is calculated explicitly for a number of lattices with low branching ratios. Two spin-spin correlation functions (far from the boundary) are also calculated explicitly and are generally shown to be continuous with discontinuous first derivatives at Tc. A possible experiment for measuring this Cayley tree behaviour is discussed.