×

On directed polymers in a random environment. (English) Zbl 0773.60067

Applied probability, Proc. Symp., Sheffield/UK 1990, IMS Lect. Notes, Monogr. Ser. 18, 41-47 (1991).
[For the entire collection see Zbl 0752.00055.]
A random walk \(\xi_ 0=0\), \(\xi_ 1,\xi_ 2,\dots\) with values in \(\mathbb{Z}^ d\) is perturbed by considering a different governing probability measure for \(\xi_ 1,\dots,\xi_ T\) defined by a likelihood ratio of the form \(\prod^ T_{s=1}X(\xi_ s,s)\) where the \(X(i,t)\) \((i\in\mathbb{Z}^ d,\;s=1,2,\dots)\) are i.i.d. with mean one. It is shown that the mean square displacement of the perturbed random walk is of the order of magnitude \(T\) provided \(\text{Var}(X)\) is small enough.

MSC:

60G50 Sums of independent random variables; random walks
82D60 Statistical mechanics of polymers

Citations:

Zbl 0752.00055
PDF BibTeX XML Cite