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 tree-like structures. Generalizations to the case where the underlying tree is irregular
in nature are also considered.
University of Utah
| Department of Mathematics
|
bressloff@math.utah.edu
Aug 2001.