Asymptotic of the maximal displacement in a branching random walk

Bastien Mallein (1)

1 DMA - Département de Mathématiques et Applications - ENS Paris

In this article, we study the maximal displacement in a branching random walk. We prove that its asymptotic behaviour consists in a first almost sure ballistic term, a negative logarithmic correction in probability and stochastically bounded fluctuations. This result, proved in [14] and [2] is given here under close-to-optimal integrability conditions. Borrowing ideas from [5] and [23], we provide simple proofs for this result, also deducing the genealogical structure of the individuals that are close to the maximal displacement.

Mots clés: branching random walk; random walk

Domaines: Probabilités [math.PR]

Bastien Mallein. Asymptotic of the maximal displacement in a branching random walk. The Graduate Journal of Mathematics, 2016, 1 (2), pp.92--104. ⟨hal-01322479⟩