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

TL;DR

This paper demonstrates that trajectories of individuals chosen via supercritical Gibbs measure in a branching random walk converge to a Brownian excursion, extending previous results and addressing questions about trajectory splitting times.

Contribution

It proves the convergence of trajectories under supercritical Gibbs measure to a Brownian excursion and refines understanding of split times between two such trajectories.

Findings

Trajectories under supercritical Gibbs measure converge to Brownian excursion.

The probability of two trajectories splitting before time t is characterized.

Extends previous convergence results to supercritical regimes.

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.