# Localization of a two-dimensional random walk with an attractive path interaction

Bolthausen, E (1994). Localization of a two-dimensional random walk with an attractive path interaction. The Annals of Probability, 22(2):875-918.

## 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.

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.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Self-attracting random walk; localization; large deviations English 1994 20 May 2010 14:57 05 Apr 2016 13:28 Institute of Mathematical Statistics 0091-1798 https://doi.org/10.1214/aop/1176988734 http://www.ams.org/mathscinet-getitem?mr=1288136http://www.zentralblatt-math.org/zmath/en/search/?q=an:0819.60028
Permanent URL: https://doi.org/10.5167/uzh-22619