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August 2020 Galton–Watson processes in varying environment and accessibility percolation
Daniela Bertacchi, Pablo M. Rodriguez, Fabio Zucca
Braz. J. Probab. Stat. 34(3): 613-628 (August 2020). DOI: 10.1214/19-BJPS434

Abstract

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 Nowak and Krug (Europhysics Letters 101 (2013) 66004).

Citation

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Daniela Bertacchi. Pablo M. Rodriguez. Fabio Zucca. "Galton–Watson processes in varying environment and accessibility percolation." Braz. J. Probab. Stat. 34 (3) 613 - 628, August 2020. https://doi.org/10.1214/19-BJPS434

Information

Received: 1 March 2018; Accepted: 1 February 2019; Published: August 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07232915
MathSciNet: MR4124543
Digital Object Identifier: 10.1214/19-BJPS434

Rights: Copyright © 2020 Brazilian Statistical Association

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Vol.34 • No. 3 • August 2020
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