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

Abstract

Considering a critical branching random walk on the real line. From a study of 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 a recent paper of A\"id\'ekon and Shi in which they solved the problem of the normalisation of the partition function in the critical regime.