  We discuss the generic slowing down of quantum dynamics in low energy density
states of spatially local Hamiltonians. Beginning with quantum walks of a
single particle, we prove that for certain classes of Hamiltonians
(deformations of lattice-regularized $H\propto p^{2k}$), the ``butterfly
velocity" of particle motion at low energies has an upper bound that must scale
as $E^{(2k-1)/2k}$, as expected from dimensional analysis. We generalize these
results to obtain bounds on the typical velocities of particles in many-body
systems with repulsive interactions, where for certain families of Hubbard-like
models we obtain similar scaling.
