  Strongly disordered and strongly interacting quantum critical points are
difficult to access with conventional field theoretic methods. They are,
however, both experimentally important and theoretically interesting. In
particular, they are expected to realize universal incoherent transport. Such
disordered quantum critical theories have recently been constructed
holographically by deforming a CFT by marginally relevant disorder. In this
paper we find additional disordered fixed points via relevant disordered
deformations of a holographic CFT. Using recently developed methods in
holographic transport, we characterize the thermal conductivity in both sets of
theories in 1+1 dimensions. The thermal conductivity is found to tend to a
constant at low temperatures in one class of fixed points, and to scale as
$T^{0.3}$ in the other. Furthermore, in all cases the thermal conductivity
exhibits discrete scale invariance, with logarithmic in temperature
oscillations superimposed on the low temperature scaling behavior. At no point
do we use the replica trick.
