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

Xinxin Chen (1) , Thomas Madaule (2) , Bastien Mallein (3)

1 PSPM - Probabilités, statistique, physique mathématique
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 DMA - Département de Mathématiques et Applications - ENS Paris

Consider a branching random walk on the real line. In a recent article, Chen [9] proved that the renormalised trajectory leading to the leftmost individual at time n converges in law to a standard Brownian excursion. Besides, in [22], 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 tra-jectory of two individuals selected according to the Gibbs measure to have split before time t, partially answering a question of [10].

Mots clés: branching random walk; Gibbs measure; random walk

Domaines: Probabilités [math.PR]

Xinxin Chen, Thomas Madaule, Bastien Mallein. On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk. Stochastic Processes and their Applications, 2018, ⟨10.1016/j.spa.2018.11.006⟩. ⟨hal-01322499⟩