  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 by Addario-Berry and
Reed, and Hu and Shi is given here under close-to-optimal integrability
conditions. We provide simple proofs for this result, also deducing the
genealogical structure of the individuals that are close to the maximal
displacement.

Probability