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Classical localization and percolation in random environments on trees

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journal contribution
posted on 11.05.2006, 16:26 by P.C. Bressloff, Vincent Dwyer, Michael J. Kearney
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.

History

School

  • Mechanical, Electrical and Manufacturing Engineering

Pages

244481 bytes

Citation

BRESSLOFF, P.C., DWYER, V.M. and KEARNEY, M.J., 1997. Classical localization and percolation in random environments on trees. Physical Review E, 55(6), pp. 6765–6775

Publisher

© American Physical Society

Publication date

1997

Notes

This article was pubished in the journal, Physical Review E [© American Physical Society]. It is also available at: http://link.aps.org/abstract/PRE/v55/p6765.

ISSN

1063-651x

Language

en

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