  We consider the dynamics of rotational excitations placed on a single
molecule in spatially disordered 1D, 2D and 3D ensembles of ultracold molecules
trapped in optical lattices. The disorder arises from incomplete populations of
optical lattices with molecules. This leads to a model corresponding to a
quantum particle with long-range tunnelling amplitudes moving on a lattice with
the same on-site energy but with forbidden access to random sites (vacancies).
We examine the time and length scales of Anderson localization for this type of
disorder with realistic experimental parameters in the Hamiltonian. We show
that for an experimentally realized system of KRb molecules on an optical
lattice this type of disorder leads to disorder-induced localization in 1D and
2D systems on a time scale $t \sim 1$ sec. For 3D lattices with $55$ sites in
each dimension and vacancy concentration $ 90~\%$, the rotational excitations
diffuse to the edges of the lattice and show no signature of Anderson
localization. We examine the role of the long-range tunnelling amplitudes
allowing for transfer of rotational excitations between distant lattice sites.
Our results show that the long-range tunnelling has little impact on the
dynamics in the diffusive regime but affects significantly the localization
dynamics in lattices with large concentrations of vacancies, enhancing the
width of the localized distributions in 2D lattices by more than a factor of 2.
Our results raise a general question whether quantum particles with long-range
tunnelling can undergo quantum localization in 3D lattices with substitutional
disorder.
