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.
Publié le : 1994-04-14
Classification:
Self-attracting random walk,
localization,
large deviations,
60K35,
60F10,
60J25
@article{1176988734,
author = {Bolthausen, Erwin},
title = {Localization of a Two-Dimensional Random Walk with an Attractive Path Interaction},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 875-918},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988734}
}
Bolthausen, Erwin. Localization of a Two-Dimensional Random Walk with an Attractive Path Interaction. Ann. Probab., Tome 22 (1994) no. 4, pp. 875-918. http://gdmltest.u-ga.fr/item/1176988734/