GRIFFITHS SINGULARITIES IN DILUTED ISING-MODELS ON THE CAYLEY TREE

The Griffiths singularities are fully exhibited for a class of diluted ferromagnetic Ising models defined on the Cayley tree (Bethe lattice) . For the deterministic model the Lee-Yang circle theorem is explicitl y proven for the magnetization at the origin and it is shown that, in the thermodynamic limit, the Lee-Yang singularities become dense in th e entire unit circle for the whole ferromagnetic phase. Smoothness (in finite differentiability) of the quenched magnetization m at the origi n with respect to the external magnetic field is also proven for conve nient choices of temperature and disorder. From our analysis we also c onclude that the existence of metastable states is impossible for the random models under consideration.