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

Bolthausen, Erwin

Commun. Math. Phys. 123, No. 4, 529-534 (1989).

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

MSC:

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)

Keywords:

random walk in random scenery; directed polymers; central limit theorem; martingale arguments

References:

[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