A note on the diffusion of directed polymers in a random environment
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.
scientific article; zbMATH DE number 4121121