  We study transport in a one-dimensional boundary-driven Anderson insulator
(the XX spin chain with onsite disorder) with randomly positioned onsite
dephasing, observing a transition from diffusive to subdiffusive spin transport
below a critical density of sites with dephasing. This model is intended to
mimic the passage of an excitation through (many-body) insulating regions or
ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion
transition observed in the disordered Heisenberg model [1]. We also present the
exact solution of a semiclassical model of conductors and insulators introduced
in Ref. 2, which exhibits both diffusive and subdiffusive phases, and
qualitatively reproduces the results of the quantum system. The critical
properties of both models, when passing from diffusion to subdiffusion, are
interpreted in terms of "Griffiths effects". We show that the finite-size
scaling comes from the interplay of three characteristic lengths: one
associated with disorder (the localization length), one with dephasing, and the
third with the percolation problem defining large, rare, insulating regions. We
conjecture that the latter, which grows logarithmically with system size, may
potentially be responsible for the fact that heavy-tailed resistance
distributions typical of Griffiths effects have not been observed in
subdiffusive interacting systems.
