LOCALIZATION OF A 2-DIMENSIONAL RANDOM-WALK WITH AN ATTRACTIVE PATH INTERACTION

We consider an ordinary, symmetric, continuous-time random walk on the two-dimensional lattice Z2. 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 --> infinity. It turns out that the displacement is asymptotically of order T1/4. The main technique for proving the result is a refined analysis of large deviation probabilities. A partial discussion is given also for higher dimensions.