## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60F10: Large deviations 60J25: Continuous-time Markov processes on general state spaces

**Keywords**

Self-attracting random walk localization large deviations

#### Citation

Bolthausen, Erwin. Localization of a Two-Dimensional Random Walk with an Attractive Path Interaction. Ann. Probab. 22 (1994), no. 2, 875--918. doi:10.1214/aop/1176988734. https://projecteuclid.org/euclid.aop/1176988734