2134/1720 P.C. Bressloff P.C. Bressloff Vincent Dwyer Vincent Dwyer Michael J. Kearney Michael J. Kearney Classical localization and percolation in random environments on trees Loughborough University 2006 untagged Mechanical Engineering not elsewhere classified 2006-05-11 16:26:47 Journal contribution https://repository.lboro.ac.uk/articles/journal_contribution/Classical_localization_and_percolation_in_random_environments_on_trees/9385706 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.