Classical localization and percolation in random environments on trees

Paul C. Bressloff, Vincent M. Dwyer, Michael J. Kearney
1997 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
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 ͑1996͔͒, exact criteria
more » ... 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.
doi:10.1103/physreve.55.6765 fatcat:7gdmoc4e2zhkdmz26z5o7ttg5u