Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T09:15:11.232Z Has data issue: false hasContentIssue false
  • English
  • Français

FRACTAL AND RESISTANCE DIMENSIONS OF RANDOM TREES

Published online by Cambridge University Press:  22 October 2007

Mokhtar H. Konsowa  and
Reem A. Al-Jarallah
Mokhtar H. Konsowa
Affiliation:
Department of Statistics and Operations Research Faculty of Science, University of Kuwait, Safat 13060Kuwait E-mail: [email protected]; [email protected]
Reem A. Al-Jarallah
Affiliation:
Department of Statistics and Operations Research Faculty of Science, University of Kuwait, Safat 13060Kuwait E-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article we determine a formula for fractal and resistance dimensions of two models of uniformly bounded random trees. The type (transient or recurrent) of the random walk on such trees is ascribed, to some extent, to these dimensions. The results presented in this article generalize some of the results of [6] and [7].

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 21 , Issue 4 , October 2007 , pp. 623 - 634
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Berman, K. & Konsowa, M. (1989). Random paths and cuts, electrical networks, and reversible Markov chains. SIAM Journal of Discrete Mathematics 3(3): 311–319.CrossRefGoogle Scholar
2.Doyle, P. & Snell, L. (1984). Random walks and electric networks. The Carus Mathematical Monographs. Washington, DC: Mathematical Association of America.Google Scholar
3.Konsowa, M. (2005). Random walks and dimensions of random trees. Submitted.Google Scholar
4.Konsowa, M.H. (2003). A note on the conductivity of random trees. Probability in the Engineering and Informational Sciences 17: 571–576.Google Scholar
5.Konsowa, M.H. & Mitro, J. (1991). The type problem for random walks on trees. Journal of Theoretical Probability 4(3): 535–550.Google Scholar
6.Konsowa, M.H. & Oraby, T.F. (2003). Dimensions of random trees. Statistics and Probability Letters 62: 49–60.Google Scholar
7.Konsowa, M.H. & Oraby, T.F. (2003). Fractal dimensions and random walks on random trees. Journal of Statistical Planning and Inference 116: 333–342.CrossRefGoogle Scholar
8.Lyons, R. (1992). Random walks, capacity, and percolation on trees. Annals of Probability 20: 2043–2088.Google Scholar
9.Lyons, R. & Peres, Y. (2004). Probability on trees and networks. Cambridge: Cambridge University Press. Available from http//php.indiana.edu/~rdlyons.Google Scholar
10.Lyons, T. (1983). A simple criterion for transience of a reversible Markov chain. Annals of Probability 11: 393–402.Google Scholar
11.Nash-Williams, C.st. J.A. (1959). Random walks and electric currents in networks. Proceeding of Cambridge Philosophical Society 55: 181–194.CrossRefGoogle Scholar
12.Peres, Y. (1999). Probability on trees. Lectures Notes in Mathematics Vol. 1717. Berlin: Springer, pp. 193–280.Google Scholar
13.Telcs, A. (1989). Random walks on graphs, electric networks, and fractals. Probability Theory and Related Fields 82: 435–449.CrossRefGoogle Scholar
14.Zhou, X.Y. (1993). Resistance dimension, random walk dimension, and fractal dimension. Journal of Theoretical Probability 6(4): 635–652.Google Scholar