Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2007

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

Résumé

We establish an unusual 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 [7]. 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 [14]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.
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Dates et versions

hal-00133596 , version 1 (26-02-2007)
hal-00133596 , version 2 (05-04-2007)
hal-00133596 , version 3 (03-03-2008)
hal-00133596 , version 4 (22-06-2009)

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Yueyun Hu, Zhan Shi. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees.. 2007. ⟨hal-00133596v2⟩
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