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

Abstract : 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.
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Annals of Probability, Institute of Mathematical Statistics, 2009, 37 (2), pp.742-789
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Dernière modification le : jeudi 27 avril 2017 - 09:46:10
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Yueyun Hu, Zhan Shi. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees.. Annals of Probability, Institute of Mathematical Statistics, 2009, 37 (2), pp.742-789. <hal-00133596v4>

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