  Classical and quantum conduction on a bond-diluted Bethe lattice is
considered. The bond dilution is subject to the constraint that every occupied
bond must have at least $k-1$ neighboring occupied bonds, i.e. $k$-core
diluted. In the classical case, we find the onset of conduction for $k=2$ is
continuous, while for $k=3$, the onset of conduction is discontinuous with the
geometric random first-order phase transition driving the conduction
transition. In the quantum case, treating each occupied bond as a random
scatterer, we find for $k=3$ that the random first-order phase transition in
the geometry also drives the onset of quantum conduction giving rise to a new
universality class of Anderson localization transitions.
