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 $+ϵ$ or -$ϵ$ with probability 1/2 where $ϵ$ has to be chosen small enough. Set

Then for almost all h and all $n_1,···,n_d\in \mathbb{N}$, as T tends to infinity, ${\prod }_{j=1}^d{[{\xi }_j\left(T\right)/\sqrt{T}]}^{n_j}$ and $\kappa$ (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.