CLASSICAL LOCALIZATION AND PERCOLATION IN RANDOM-ENVIRONMENTS ON TREES

We consider a simple model of transport on a regular tree, whereby spe cies evolve according to the drift-diffusion equation, and the drift v elocity on each branch of the tree is a quenched random variable. The inverse of the steady-state amplitude at the origin is expressed in te rms of a random geometric series whose convergence or otherwise determ ines whether the system is localized or delocalized. In a recent paper [P. C. Bressloff er al., Phys. Rev. Lett. 77, 5075 (1996)], exact cri teria were presented that enable one to determine the critical phase b oundary for the transition, valid for any distribution of the drift ve locities. In this paper we present a detailed derivation of these crit eria, consider a number of examples of interest, and establish a conne ction with conventional percolation theory. The latter suggests a wide r application of the results to other models of statistical processes occurring on treelike structures. Generalizations to the case where th e underlying tree is irregular in nature are also considered.