  The study of generic properties of quantum states has lead to an abundance of
insightful results. A meaningful set of states that can be efficiently prepared
in experiments are ground states of gapped local Hamiltonians, which are well
approximated by matrix product states. In this work, we prove a number of
bounds on various quantities of non-translation-invariant matrix product states
if drawn from a local i.i.d. Haar-measure. We arrive at these results by making
use of techniques for computing moments of random unitary matrices and by
exploiting a mapping to partition functions of classical statistical models, a
method that has lead to valuable insights on local random quantum circuits.
Specifically, we prove that such disordered random matrix product states
equilibrate exponentially well with overwhelming probability under the time
evolution of Hamiltonians featuring a non-degenerate spectrum. Moreover, we
prove two results about the entanglement R\'{e}nyi entropy: The entropy with
respect to sufficiently disconnected subsystems is generically extensive in the
system size, and for small connected systems the entropy is almost maximal for
sufficiently large bond dimensions.
