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.
Citation
Erwin Bolthausen. "Localization of a Two-Dimensional Random Walk with an Attractive Path Interaction." Ann. Probab. 22 (2) 875 - 918, April, 1994. https://doi.org/10.1214/aop/1176988734
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