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

Yueyun Hu; Zhan Shi

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arXiv · math/0702799

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

Yueyun Hu·Zhan Shi

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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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Yueyun Hu, Zhan Shi. 2009-06-22. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. https://arxiv.org/abs/math/0702799

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