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A note on the diffusion of directed polymers in a random environment. (English) Zbl 0684.60013
Let $\xi$ (t), $t\in {\bbfN}$, be an ordinary symmetric random walk on ${\bbfZ}\sp d$, $d>2$, 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), $t\in {\bbfN}$, $y\in {\bbfZ}\sp d$, be i.i.d. random variables which are $+\epsilon$ or -$\epsilon$ with probability 1/2 where $\epsilon$ has to be chosen small enough. Set $$ \kappa (T):=\prod\sp{T}\sb{j=1}[1+h(j,\xi (j))]. $$ Then for almost all h and all $n\sb 1,...,n\sb d\in {\bbfN}$, as T tends to infinity, $\prod\sp{d}\sb{j=1}[\xi\sb j(T)/\sqrt{T}]\sp{n\sb j}$ and $\kappa$ (T) are asymptotically uncorrelated. This generalizes a result of {\it J. Z. Imbrie} and {\it 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.
Reviewer: K.Fleischmann

60F05Central limit and other weak theorems
60G50Sums of independent random variables; random walks
82D30Random media, disordered materials (statistical mechanics)
Full Text: DOI
[1] Imbrie, J.Z., Spencer, T.: Diffusions of directed polymers in a random environment. J. Stat. Phys.52, 609 (1988) · Zbl 1084.82595 · doi:10.1007/BF01019720
[2] Neveu, J.: Discrete parameter martingales. Amsterdam: North-Holland 1975 · Zbl 0345.60026