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

Xinxin Chen, Thomas Madaule and Bastien Mallein

Stochastic Processes and their Applications, 2019, vol. 129, issue 10, 3821-3858

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).

Keywords: Branching random walk; Overlap distribution; Extremal process; Random walk; Excursion (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1016/j.spa.2018.11.006

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