Classical localization and percolation in random environments on trees

We consider a simple model of transport on a regular tree, whereby species evolve according to the drift-diffusion equation, and the drift velocity on each branch of the tree is a quenched random variable. The inverse of the steady-state amplitude at the origin is expressed in terms of a random geometric series whose convergence or otherwise determines whether the system is localized or delocalized. In a recent paper [P. C. Bressloff et al., Phys. Rev. Lett. 77, 5075 (1996)], exact criteria were presented that enable one to determine the critical phase boundary for the transition, valid for any distribution of the drift velocities. In this paper we present a detailed derivation of these criteria, consider a number of examples of interest, and establish a connection with conventional percolation theory. The latter suggests a wider application of the results to other models of statistical processes occurring on treelike structures. Generalizations to the case where the underlying tree is irregular in nature are also considered.