  We discuss symmetry breaking quantum phase transitions on the oft studied
Bethe lattice in the context of the ferromagnetic scalar spherical model or,
equivalently, the infinite $N_f$ limit of ferromagnetic models with $O(N_f)$
symmetry. We show that the approach to quantum criticality is characterized by
the vanishing of a gap to just the global modes so that {\it all} local
correlation functions continue to exhibit massive behavior. This behavior
persists into the broken symmetry phase even as the order parameter develops an
expectation value and thus there are no massless Goldstone bosons in the
spectrum. We relate this feature to a spectral property of the graph Laplacian
shared by the set of `expander' graphs, and argue that our results apply to
symmetry breaking transitions on such graphs quite generally.
