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Routeing on trees

Part of: Markov processes Special processes

Published online by Cambridge University Press:  21 June 2016

Maria Deijfen  and
Nina Gantert
Maria Deijfen*
Affiliation:
Stockholm University
Nina Gantert*
Affiliation:
Technische Universität München
*
* Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. Email address: [email protected]
** Postal address: Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany.
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Abstract

We consider three different schemes for signal routeing on a tree. The vertices of the tree represent transceivers that can transmit and receive signals, and are equipped with independent and identically distributed weights representing the strength of the transceivers. The edges of the tree are also equipped with independent and identically distributed weights, representing the costs for passing the edges. For each one of our schemes, we derive sharp conditions on the distributions of the vertex weights and the edge weights that determine when the root can transmit a signal over arbitrarily large distances.

Keywords

Treestransmissionfirst passage percolationbranching random walkMarkov chain

MSC classification

Primary: 60K37: Processes in random environments
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 475 - 488
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1]Balister, P., Bollobás, B. and Walters, M. (2009).Random transceiver networks.Adv. Appl. Prob. 41, 323–343.CrossRefGoogle Scholar
[2]Benjamini, I. and Peres, Y. (1994).Markov chains indexed by trees.Ann. Prob. 22, 219–243.Google Scholar
[3]Benjamini, I. and Peres, Y. (1994).Tree-indexed random walks on groups and first passage percolation.Prob. Theory Relat. Fields 98, 91–112.Google Scholar
[4]Biggins, J. D. (1976).The first- and last-birth problems for a multitype age-dependent branching process.Adv. Appl. Prob. 8, 446–459.Google Scholar
[5]Dembo, A. and Zeitouni, O. (1998).Large Deviations Techniques and Applications (Appl. Math. New York 38), 2nd edn.Springer, New York.Google Scholar
[6]Dousse, O. (2012).Percolation in directed random geometric graphs. In Proc. IEEE Internat. Symp. Inf. Theory, IEEE, New York, pp.601–605.Google Scholar
[7]Gantert, N., Hu, Y. and Shi, Z. (2011).Asymptotics for the survival probability in a killed branching random walk.Ann. Inst. H. Poincaré Prob. Statist. 47, 111–129.Google Scholar
[8]Hammersley, J. M. (1974).Postulates for subadditive processes.Ann. Prob. 2, 652–680.CrossRefGoogle Scholar
[9]Kingman, J. F. C. (1975).The first birth problem for an age-dependent branching process.Ann. Prob. 3, 790–801.Google Scholar
[10]Lyons, R. and Pemantle, R. (1992).Random walk in a random environment and first-passage percolation on trees.Ann. Prob. 20, 125–136. (Correction: 31 (2003), 528–529.) Google Scholar
[11]Lyons, R. and Peres, Y.Probability on trees and networks. Available at http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html.Google Scholar
[12]Mogul'skiĭ, A. A. (1977).Large deviations for trajectories of multi-dimensional random walks.Theory Prob. Appl. 21, 300–315.Google Scholar
[13]Peres, Y. (1999).Probability on trees: an introductory climb. In Lectures on Probability Theory and Statistics (Lecture Notes Math.1717), Springer, Berlin, pp.193–280.Google Scholar
[14]Shi, Z. (2011).Random walks and trees.ESAIM Proc. 31, 1–39.Google Scholar