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Survival of inhomogeneous Galton-Watson processes

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Erik Broman  and
Ronald Meester
Erik Broman*
Affiliation:
Chalmers University of Technology
Ronald Meester*
Affiliation:
VU University Amsterdam
*
∗ Postal address: Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. Email address: [email protected]
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Abstract

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We study the survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an almost sure constant. We also shed some light on the way in which the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parameterized by the retention probability p. We provide growth rates, uniformly in p, of the percolation clusters, and also show uniform convergence of the survival probability from the nth level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalizations of results in Lyons (1992).

Keywords

Inhomogeneous Galton-Watson treecontinuity of percolation functionsbranching number

MSC classification

Primary: 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 798 - 814
Copyright
Copyright © Applied Probability Trust 2008 

References

Agresti, A. (1975). On the extinction times of varying and random environment branching processes. J. Appl. Prob. 12, 39–46.Google Scholar
Broman, E. I., Häggström, O. and Steif, J. E. (2006). Refinements of stochastic domination. Prob. Theory Relat. Fields 136, 587–603.Google Scholar
Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.Google Scholar
Lyons, R. (1990). Random walks and percolation on trees. Ann. Prob. 18, 931–958.Google Scholar
Lyons, R. (1992). Random walks, capacity and percolation on trees. Ann. Prob. 20, 2043–2088.Google Scholar
Lyons, R. Probability on trees and networks. In preparation. Available at http://mypage.iu.edu/∼rdlyons/prbtree/prbtree.html.Google Scholar
Van den Berg, J. and Keane, M. (1982). On the continuity of the percolation probability function. In Conference in Modern Analysis and Probability, eds Beats, R. et al., AMS, Providence, RI, pp. 61–65.Google Scholar