Localization-delocalization transition for drift diffusion in a random environment

We investigate the localization-delocalization transition for the drift-diffusion equation on a regular tree with quenched random drift velocities on its branches. The inverse of the steady-state amplitude at the origin is expressed in terms of a random geometric series whose convergence or otherwise determines the critical phase boundary. We establish exact criteria for localization valid for an arbitrary distribution of the drift velocities. The phase transition is shown to be first order except in the percolation limit.