Slower deviations of the branching Brownian motion and of branching random walks *

We have shown recently how to calculate the large deviation function of the position X max ( t ) 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 X max ( t ) has, asymptotically in time, a prefactor characterized by a non trivial power law.