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

Abstract

Consider a branching random walk on the real line. Madaule showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen 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.

Main theorem refined, to appear in Stochastic Process. Appl

References in corpus (5)

• Invariance principles for random walks conditioned to stay positive
• Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field
• Genealogy of the extremal process of the branching random walk
• Asymptotic of the maximal displacement in a branching random walk
• On the overlap distribution of branching random walks

Cited by in corpus (2)

• Genealogy of the extremal process of the branching random walk
• Efficient approximation of branching random walk Gibbs measures