Mean-field description of Anderson localization transition

The Anderson model of noninteracting disordered electrons is studied in high spatial dimensions. In this limit the coupled Bethe-Salpeter equations determining two-particle vertices (parquet equations) reduce to a single algebraic equation for a local vertex. We find a disorder-driven bifurcation point in this equation signaling vanishing of electron diffusion and onset of Anderson localization. There is no bifurcation in $d=1,2$ where all states are localized. In dimensions $d\geq 3$ the mobility edge separating metallic and insulating phase is found for various types of disorder and compared with results of other treatments.