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Bertacchi, Daniela; Rodriguez, Pablo M.; Zucca, Fabio

Galton-Watson processes in varying environment and accessibility percolation. (English) Zbl 1446.60059

Braz. J. Probab. Stat. 34, No. 3, 613-628 (2020).
Summary: This paper deals with branching processes in varying environment with selection, where the offspring distribution depends on the generation and every particle has a random fitness which can only increase along genealogical lineages (descendants with small fitness do not survive). We view the branching process in varying environment (BPVE) as a particular example of branching random walk. We obtain conditions for the survival or extinction of a BPVE (with or without selection), using fixed point techniques for branching random walks. These conditions rely only on the first and second moments of the offspring distributions. Our results can be interpreted in terms of accessibility percolation on Galton-Watson trees. In particular, we obtain that there is no accessibility percolation on almost every Galton-Watson tree where the expected number of offspring grows sublinearly in time, while superlinear growths allows percolation. This result is in agreement with what was found for deterministic trees in [S. Nowak and J. Krug, “Accessibility percolation on \(n\)-trees”, Europhys. Lett. 101, No. 6, Article ID 66004, 6 p. (2013; doi:10.1209/0295-5075/101/66004)].

Cited in 2 Documents

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

branching process; time inhomogeneous; varying environment; fitness; selection; accessibility percolation; generating function
PDFBibTeX XMLCite
\textit{D. Bertacchi} et al., Braz. J. Probab. Stat. 34, No. 3, 613--628 (2020; Zbl 1446.60059)
Full Text: DOI arXiv Euclid

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