Let (t), , be an ordinary symmetric random walk on , , starting in 0. The trajectories of this walk will be weighted with the help of the following discrete version of a time-space white noise.
Independently of let h(t,y), , , be i.i.d. random variables which are or - with probability 1/2 where has to be chosen small enough. Set
Then for almost all h and all , as T tends to infinity, and (T) are asymptotically uncorrelated.
This generalizes a result of J. Z. Imbrie and 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.