Open Access
March 2009 Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees
Yueyun Hu, Zhan Shi
Ann. Probab. 37(2): 742-789 (March 2009). DOI: 10.1214/08-AOP419

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.

Citation

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Yueyun Hu. Zhan Shi. "Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees." Ann. Probab. 37 (2) 742 - 789, March 2009. https://doi.org/10.1214/08-AOP419

Information

Published: March 2009
First available in Project Euclid: 30 April 2009

zbMATH: 1169.60021
MathSciNet: MR2510023
Digital Object Identifier: 10.1214/08-AOP419

Subjects:
Primary: 60J80

Keywords: Branching random walk , directed polymer on a tree , marked tree , martingale convergence , Minimal position , Spine

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 2 • March 2009
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