First order transition for the branching random walk at the critical parameter

Consider a branching random walk on the real line in the boundary case. The associated additive martingales can be viewed as the partition function of a directed polymers on a disordered tree. By studying the law of the trajectory of a particle chosen under the polymer measure, we establish a first order transition for the partition function at the critical parameter. This result is strongly related to the paper of Aïdékon and Shi (2014) in which they solved the problem of the normalization of the partition function in the critical regime.