  The study of generic properties of quantum states has led 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 introduce a picture of
generic states within the trivial phase of matter with respect to their
non-equilibrium and entropic properties: We do so by rigorously exploring
non-translation-invariant matrix product states drawn from a local i.i.d.
Haar-measure. We arrive at these results by exploiting 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
Renyi 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.
