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September 2021 To fixate or not to fixate in two-type annihilating branching random walks
Daniel Ahlberg, Simon Griffiths, Svante Janson
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Ann. Probab. 49(5): 2637-2667 (September 2021). DOI: 10.1214/21-AOP1521

Abstract

We study a model of competition between two types evolving as branching random walks on Zd. The two types are represented by red and blue balls, respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of Zd contain one ball each which are independently coloured red with probability p and blue otherwise. We address the question of fixation, referring to the sites and eventually settling for a given colour or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for p1/2 and that every site will change colour infinitely often almost surely for the balanced initial condition p=1/2.

Funding Statement

This work was supported in part by Grant 2016-04442 from the Swedish Research Council (DA); CNPq bolsa de produtividade Proc. 310656/2016-8 and FAPERJ Jovem cientista do nosso estado Proc. 202.713/2018 (SG); the Knut and Alice Wallenberg Foundation, the Isaac Newton Institute for Mathematical Sciences (EPSRC Grant Number EP/K032208/1) and the Simons foundation (SJ).

Acknowledgements

The authors are very grateful to Robert Morris, for his encouragement to pursue this project and his valuable input in several joint discussions. The authors are also grateful to Luca Avena and Conrado da Costa for informing them about the work of Pruitt [26] (cf. Remark 3.3), to IMPA and to the Isaac Newton Institute, where parts of this work were done, and to an anonymous referee who found a gap in the argument.

Citation

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Daniel Ahlberg. Simon Griffiths. Svante Janson. "To fixate or not to fixate in two-type annihilating branching random walks." Ann. Probab. 49 (5) 2637 - 2667, September 2021. https://doi.org/10.1214/21-AOP1521

Information

Received: 1 February 2020; Revised: 1 February 2021; Published: September 2021
First available in Project Euclid: 24 September 2021

Digital Object Identifier: 10.1214/21-AOP1521

Subjects:
Primary: 60J80 , 60K35 , 82C22 , 82C27

Keywords: Branching random walk , competing growth , nonequilibrium dynamics

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 5 • September 2021
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