Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees.

Yueyun Hu (1) , Zhan Shi (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 [9]. 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 [17]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

Mots clés: directed polymer on a tree; Branching random walk; minimal position; martingale convergence; spine; marked tree; directed polymer on a tree.

Domaines: Probabilités [math.PR]

Yueyun Hu, Zhan Shi. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees.. 2008. ⟨hal-00133596v3⟩