We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609–631]. Our method applies, furthermore, to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [J. Statist. Phys. 51 (1988) 817–840]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.
"Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees." Ann. Probab. 37 (2) 742 - 789, March 2009. https://doi.org/10.1214/08-AOP419