Slower deviations of the branching Brownian motion and of branching random walks**Dedicated to John Cardy on the occasion of his 70th birthday.

Abstract

We have shown recently how to calculate the large deviation function of the position of the rightmost particle of a branching Brownian motion at time t. This large deviation function exhibits a phase transition at a certain negative velocity. Here we extend this result to more general branching random walks and show that the probability distribution of has, asymptotically in time, a prefactor characterized by a non trivial power law.