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

Xinxin Chen, Thomas Madaule, Bastien Mallein

Published 2015-07-16Version 1

Consider a branching random walk on the real line. In a recent article, Chen proved that the renormalised trajectory leading to the leftmost individual at time $n$ converges in law to a standard Brownian excursion. Besides Madaule showed the renormalised trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. In this article, we prove that trajectory of individuals selected independently according to a supercritical Gibbs measure converge in law to Brownian excursions. Refinements of this results also enables to express the probability for the trajectory of two individuals selected according to the Gibbs measure to have split before time $t$, partially answering a question of Derrida and Spohn.