  The low-frequency response of systems near a many-body localization
transition can be dominated by rare regions that are locally critical or "in
the other phase". It is known that, in one dimension, these rare regions can
cause the d.c. conductivity and diffusion constant to vanish even inside the
delocalized thermal phase. Here, we present a general analysis of such
Griffiths effects in the thermal phase near the many-body localization
transition: we consider both one-dimensional and higher-dimensional systems,
subject to quenched randomness, and discuss both linear response (including the
frequency- and wavevector-dependent conductivity) and more general dynamics. In
all the regimes we consider, we identify observables that are dominated by
rare-region effects. In some cases (one-dimensional systems and Floquet systems
with no extensive conserved quantities), essentially all long-time local
observables are dominated by rare-region effects; in others, generic
observables are instead dominated by hydrodynamic long-time tails throughout
the thermal phase, and one must look at specific probes, such as spin echo, to
see Griffiths behavior.
