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On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk

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  • Chen, Xinxin
  • Madaule, Thomas
  • Mallein, Bastien

Abstract

Consider a branching random walk on the real line. Madaule (2016) showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen (2015) proved that the renormalized trajectory leading to the leftmost individual at time n converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probability for the trajectories of two individuals selected according to the Gibbs measure to have split before time t, partially answering a question of Derrida and Spohn (1988).

Suggested Citation

  • Chen, Xinxin & Madaule, Thomas & Mallein, Bastien, 2019. "On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3821-3858.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:10:p:3821-3858
    DOI: 10.1016/j.spa.2018.11.006
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    1. Madaule, Thomas, 2016. "First order transition for the branching random walk at the critical parameter," Stochastic Processes and their Applications, Elsevier, vol. 126(2), pages 470-502.
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