A note on the diffusion of directed polymers in a random environment. (English) Zbl 0684.60013

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 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.
Reviewer: K.Fleischmann


60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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
[2] Neveu, J.: Discrete parameter martingales. Amsterdam: North-Holland 1975 · Zbl 0345.60026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.