Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees
نویسندگان
چکیده
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.
منابع مشابه
Minimal Position and and Critical Martingale Convergence in Branching Random Walks
For a branching random walk on the line, an unusual almost sure limit theorem for its minimal position is proved along with a Heyde-Seneta-type convergence result for the critical martingale. The latter answers a question posed by Biggins and Kyprianou [12]. Moreover, the employed method applies to the study of directed polymers on a disordered tree and provides a rigorous proof of a phase tran...
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