Heuristics:

One can start the heuristics as in the short range case. 
The only difference comes up when estimating


Because of slow decay, one has



Because of the averaging of the random variables, this probability is not given by 
usual large deviation estimate. 

Locally the random potential is an average of r.v.; this averaging procedure is 
stronger than the one induced by the momentum localization.

The short range case = the quantum case (the momentum plays a role).

The long range case = the classical case (the momentum plays no role).