The Annals of Probability

Localization of a Two-Dimensional Random Walk with an Attractive Path Interaction

Erwin Bolthausen

Abstract

We consider an ordinary, symmetric, continuous-time random walk on the two-dimensional lattice $\mathbb{Z}^2$. The distribution of the walk is transformed by a density which discounts exponentially the number of points visited up to time $T$. This introduces a self-attracting interaction of the paths. We study the asymptotic behavior for $T \rightarrow \infty$. It turns out that the displacement is asymptotically of order $T^{1/4}$. The main technique for proving the result is a refined analysis of large deviation probabilities. A partial discussion is given also for higher dimensions.

Article information

Source
Ann. Probab., Volume 22, Number 2 (1994), 875-918.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988734

Digital Object Identifier
doi:10.1214/aop/1176988734

Mathematical Reviews number (MathSciNet)
MR1288136

Zentralblatt MATH identifier
0819.60028

JSTOR