Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-19T23:56:32.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Branching and tree indexed random walks on fractals

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

András Telcs  and
Nicholas C. Wormald
András Telcs*
Affiliation:
International Business School, Budapest
Nicholas C. Wormald*
Affiliation:
University of Melbourne
*
∗Postal address: Head of Library and Computing, International Business School, Tarogato u. 2–4, 1021 Budapest, Hungary. Partial support from OTKA T 016237 and the Australian Research Council.
∗∗Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3052, Australia. Email address: nick@unimelb.edu.au.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with the recurrence of branching random walks on polynomially growing graphs. Amongst other things, we demonstrate the strong recurrence of tree indexed random walks determined by the resistance properties of spherically symmetric graphs. Several branching walk models are considered to show how the branching mechanism influences the recurrence behaviour.

Keywords

Branching processrandom walkbranching random walkfractal

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 999 - 1011
Copyright
Copyright © Applied Probability Trust 1999 

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.)

Footnotes

Supported by the Australian Research Council.

References

Barlow, M. T. (1990). Random walks and diffusion on fractals. In Proc. International Congress of Mathematicians, ed. Satake, I. Math. Soc. Japan, Tokyo, pp. 1025–1035.Google Scholar
Barlow, M. T. (1999). Diffusion on Fractals. Preprint.Google Scholar
Benjamini, I., and Peres, Y. (1994). Tree indexed random walks on groups and first passage percolation. Prob. Theory Rel. Fields 98, 91–112.CrossRefGoogle Scholar
Benjamini, I., and Peres, Y. (1994). Markov chains indexed by trees. Ann. Prob. 22, 219–243.CrossRefGoogle Scholar
Billingsley, P. (1986). Probability and Measure, 2nd edn. Wiley, New York.Google Scholar
Doyle, P. G., and Snell, J. L. (1984). Random Walks and Electric Networks (Carus Math. Monog. 22). MAA, Washington.Google Scholar
Durrett, R. (1996). Probability Theory and Examples. Duxbury Press, Belmont, CA. Wadsworth and Brooks.Google Scholar
Hambley, B. (1991). On the limiting distribution of a supercritical branching process on a random environment. J. Appl. Prob. 29, 499–518.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, New York.Google Scholar
Klebaner, F. C. (1989). Stochastic difference equations and generalised gamma distribution. Ann. Prob. 17, 178–188.Google Scholar
Rammal, R., and Toulouse, G. (1983). Random walks on fractal structures and percolation clusters. J. Phys. Lett. 44, 13–22.Google Scholar
Schinazi, R. (1993). On multiple phase transitions for branching Markov chains. J. Statist. Phys. 71, 507–511.Google Scholar
Telcs, A. (1989). Random walks on graphs, electric networks and fractals. Prob. Theory Rel. Fields 82, 435–449.Google Scholar
Telcs, A. (1995). Spectra of graphs and fractal dimensions II. J. Theoret. Prob. 8, 77–96.CrossRefGoogle Scholar
Telcs, A. (1999). Fractal dimensions and Martin boundary of graphs. Submitted to Studia Sci. Math. Hungarica.Google Scholar