  We prove that generic quantum local Hamiltonians are gapless. In fact, we
prove that there is a continuous density of states above the ground state. The
Hamiltonian can be on a lattice in any spatial dimension or on a graph with a
bounded maximum vertex degree. The type of interactions allowed for include
translational invariance in a disorder (i.e., probabilistic) sense with some
assumptions on the local distributions. Examples include many-body localization
and random spin models. We calculate the scaling of the gap with the system's
size when the local terms are distributed according to a Gaussian
$\beta-$orthogonal random matrix ensemble. As a corollary there exist finite
size partitions with respect to which the ground state is arbitrarily close to
a product state. When the local eigenvalue distribution is discrete, in
addition to the lack of an energy gap in the limit, we prove that the ground
state has finite size degeneracies. The proofs are simple and constructive.
This work excludes the important class of truly translationally invariant
Hamiltonians where the local terms are all equal.
