  We use the recently developed finite cluster typical medium approach to study
the Anderson localization transition in three dimensions. Applying our method
to the box and binary alloy disorder distributions, we find a fast convergence
with the cluster size. We demonstrate the importance of the typical medium
environment and the non-local spatial correlations for the proper
characterization of the localization transition. As the cluster size increases,
our typical medium cluster method recovers the correct critical disorder
strength for the transition. Our findings highlight the importance of the
non-local cluster corrections for capturing the localization behavior of the
mobility edge trajectories. Our results demonstrate that the typical medium
cluster approach developed here provides a consistent and systematic
description of the Anderson localization transition in the framework of the
effective medium embedding schemes.
