SURVIVAL OF INHOMOGENEOUS GALTON-WATSON PROCESSES
Journal article, 2008

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 front 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).

branching number

Inhomogeneous Galton-Watson tree

TREES

PERCOLATION

continuity of percolation functions

RANDOM-WALKS

Author

Erik Broman

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

R. Meester

Free University of Amsterdam

Advances in Applied Probability

0001-8678 (ISSN) 1475-6064 (eISSN)

Vol. 40 3 798-814

Subject Categories

Mathematics

DOI

10.1239/aap/1222868186

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Latest update

3/6/2018 9