  By tightening the conventional Lieb-Robinson bounds to better handle systems
which lack translation invariance, we determine the extent to which "weak
links" suppress operator growth in disordered one-dimensional spin chains. In
particular, we prove that ballistic growth is impossible when the distribution
of coupling strengths $\mu(J)$ has a sufficiently heavy tail at small $J$, and
identify the correct dynamical exponent to use instead. Furthermore, through a
detailed analysis of the special case in which the couplings are genuinely
random and independent, we find that the standard formulation of Lieb-Robinson
bounds is insufficient to capture the complexity of the dynamics -- we must
distinguish between bounds which hold for all sites of the chain and bounds
which hold for a subsequence of sites, and we show by explicit example that
these two can have dramatically different behaviors. All the same, our result
for the dynamical exponent is tight, in that we prove by counterexample that
there cannot exist any Lieb-Robinson bound with a smaller exponent. We close by
discussing the implications of our results, both major and minor, for numerous
applications ranging from quench dynamics to the structure of ground states.
