02-389 Francois Germinet, Abel Klein

The Anderson metal-insulator transport transition (370K, .ps) Sep 19, 02

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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.

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