The Annals of Probability

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

Yueyun Hu and 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 [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.

Article information

Source
Ann. Probab., Volume 37, Number 2 (2009), 742-789.

Dates
First available in Project Euclid: 30 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1241099928

Digital Object Identifier
doi:10.1214/08-AOP419

Mathematical Reviews number (MathSciNet)
MR2510023

Zentralblatt MATH identifier
1169.60021

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching random walk minimal position martingale convergence spine marked tree directed polymer on a tree

Citation

Hu, Yueyun; Shi, Zhan. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 (2009), no. 2, 742--789. doi:10.1214/08-AOP419. https://projecteuclid.org/euclid.aop/1241099928


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