Abstract
We study a model of competition between two types evolving as branching random walks on . 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 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 and that every site will change colour infinitely often almost surely for the balanced initial condition .
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
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
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