Francois Germinet, Abel Klein
The Anderson metal-insulator transport transition
(370K, .ps)

ABSTRACT.  We discuss a new approach to the metal-insulator 
transition for random operators, based on transport instead of 
spectral properties. It applies to random Schr\"odinger operators, 
acoustic operators in random media, and Maxwell operators in 
random media. We define a 
 local transport exponent $\beta(E)$, and set the \emph{metallic 
transport region} to be the part of the spectrum with nontrivial 
transport (i.e., $\beta(E)>0$). The \emph{strong insulator region} 
 is taken to be the part of the 
 spectrum where the random operator exhibits strong dynamical 
localization in the Hilbert-Schmidt norm, and hence no transport. 
For the standard random operators satisfying a Wegner estimate, 
 these metallic and insulator regions 
are shown to be complementary sets in the spectrum of the random 
operator, and the local transport exponent $\beta(E)$ provides a 
characterization of the \emph{metal-insulator transport transition}. 
If such a transition occurs, then $\beta(E)$ has to be discontinuous 
at a \emph{transport mobility edge}: if the transport is nontrivial 
then $\beta(E)\ge \frac 1{2bd}$, where $d$ is the space dimension and 
$b\ge 1$ is the power of the volume in Wegner's estimate. 
We also examine the transport transition for random 
polymer models, where the random dimer models provide 
 explicit examples of the transport transition and of a 
transport mobility edge.