  We study the diluted Ising ferromagnet on the Bethe lattice as a case study
for the application of the cavity method to problems with Griffiths-McCoy
singularities. Specifically, we are able to make much progress at infinite
coupling where we compute, from the cavity method, the density of Lee-Yang
zeroes in the paramagnetic Griffiths region as well as the properties of the
phase transition to the ferromagnet. This phase transition is itself of a
Griffiths-McCoy character albeit with a power law distribution of cluster
sizes.
