### Minimum of a branching random walk

ABSTRACT

Take one particle located at zero on the real line.

At time 1, this particle dies and gives birth to a point process.

Then at each time n, the particles die and give birth to independent copies of the same point process, translated around their position.

We investigate the minimum at time n of this process, that we call M_{n}. The first order M_{n}/n is well known and we suppose by renormalization that M_{n}/n goes to zero.

The second order was recently found separately by Hu, Shi (2009) and Addario-Berry,Reed (2009), where it is shown that M_{n}~ (3/2) log n. We show that a convergence in law holds for the recentered minimum M_{n} - (3/2) log n.