Absence of transport in Anderson localization

We consider the charge transport in the tight-binding Anderson model. Under a mild condition on the Fermi projection, we show that it is zero almost surely. This result has wider applicability than our previous work [12], while the definition of charge transport is slightly different. It also applies to the computation of non-diagonal component of the conductivity tensor which recovers the famous result of quantization of Hall conductivity in quantum Hall systems.