CLASSICAL LOCALIZATION FOR THE DRIFT-DIFFUSION EQUATION ON A CAYLEY TREE

We show that classical localization occurs for the drift-diffusion equ ation on an ordered Cayley tree when the drift velocity v on each bran ch of the tree exceeds a critical value v(c) = DL(-1) In(z-1), where z is the coordination number, D is the diffusion constant and L is the segment length. For v < v(c) the asymptotic decay of the delocalized s tate exhibits conventional diffusive behaviour, whereas at the critica l point v = v(c) there is anomalous behaviour in the form of a critica l slowing-down. A necessary condition for localization in the presence of randomly distributed drift velocities is also derived.