Classical localization and percolation in random environments on trees
File(s)
Accepted version
Author(s)
Bressloff, Paul C
Dwyer, Vincent M
Kearney, Michael J
Type
Journal Article
Abstract
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.
Date Issued
1997-06
Date Acceptance
1997-06-01
Citation
Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 1997, 55 (6), pp.6765-6775
ISSN
1539-3755
Publisher
American Physical Society
Start Page
6765
End Page
6775
Journal / Book Title
Physical Review E: Statistical, Nonlinear, and Soft Matter Physics
Volume
55
Issue
6
Copyright Statement
©1997 American Physical Society
Bressloff, Paul C., Vincent M. Dwyer, and Michael J. Kearney. "Classical localization and percolation in random environments on trees." Physical Review E 55.6 (1997): 6765.
Bressloff, Paul C., Vincent M. Dwyer, and Michael J. Kearney. "Classical localization and percolation in random environments on trees." Physical Review E 55.6 (1997): 6765.
Identifier
http://dx.doi.org/10.1103/physreve.55.6765
Publication Status
Published
Article Number
6765
Date Publish Online
1997-06-01