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Zurich Open Repository and Archive

Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-22578

# Bolthausen, E; Schmock, U (1995). On self-attracting random walks. In: Cranston, M C; Pinsky, M A. Stochastic analysis (Ithaca, NY, 1993). Providence, RI, 23-44. ISBN 0-8218-0289-5.

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## Abstract

In this survey paper we mainly discuss the results contained in two of our recent articles [2] and [5]. Let {Xt}t≥0 be a continuous-time, symmetric, nearest-neighbour random walk on Zd. For every T > 0 we define the transformed
path measure dPT = (1/ZT ) exp(HT ) dP, where P is the original one and ZT is the appropriate normalizing constant. The Hamiltonian HT imparts the self-attracting
interaction of the paths up to time T. We consider the case where HT is given by a potential function V on Zd with finite support, and the case HT = −NT , where NT denotes the number of points visited by the random walk up to time T. In both cases the typical paths under PT as T →∞ clump together much more than those of the free random walk and give rise to localization phenomena.