A note on the diffusion of directed polymers in a random environment
type: Item
instance of: scholarly article
title: A note on the diffusion of directed polymers in a random environment
zbMATH Open document ID: 0684.60013
DOI: 10.1007/BF01218584
author: Erwin Bolthausen
published in: Communications in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: Let \(\xi\) (t), \(t\in {\mathbb{N}}\), be an ordinary symmetric random walk on \({\mathbb{Z}}^ d\), \(d>2\), starting in 0. The trajectories of this walk \(\xi\) will be weighted with the help of the following discrete version of a time-space white noise. Independently of \(\xi\) let h(t,y), \(t\in {\mathbb{N}}\), \(y\in {\mathbb{Z}}^ d\), be i.i.d. random variables which are \(+\epsilon\) or -\(\epsilon\) with probability 1/2 where \(\epsilon\) has to be chosen small enough. Set \[ \kappa (T):=\prod^{T}_{j=1}[1+h(j,\xi (j))]. \] Then for almost all h and all \(n_ 1,...,n_ d\in {\mathbb{N}}\), as T tends to infinity, \(\prod^{d}_{j=1}[\xi_ j(T)/\sqrt{T}]^{n_ j}\) and \(\kappa\) (T) are asymptotically uncorrelated. This generalizes a result of \textit{J. Z. Imbrie} and \textit{T. Spencer} [Diffusion of directed polymers in a random environment, J. Stat. Phys. 52, pp. 609 (1988)] and implies a central limit theorem. Simple martingale arguments are used in the proofs.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 4121121
zbMATH Keywords: random walk in random scenery
zbMATH Keywords: directed polymers
zbMATH Keywords: central limit theorem
zbMATH Keywords: martingale arguments
MaRDI profile type: Publication
cites work: Diffusion of directed polymers in a random environment.
cites work: Q4114574
full work available at URL: https://doi.org/10.1007/bf01218584
OpenAlex ID: W2057763268
description: scientific article; zbMATH DE number 4121121