## On self-attracting random walks.(English)Zbl 0829.60021

Cranston, Michael C. (ed.) et al., Stochastic analysis. Proceedings of the Summer Research Institute on stochastic analysis, held at Cornell University, Ithaca, NY, USA, July 11-30, 1993. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 57, 23-44 (1995).
Let $$((X_t, t \geq 0), P)$$ denote a continuous-time, symmetric, nearest-neighbor random walk on $$Z^d$$. For every $$T>0$$ define the transformed path measure $$dP_T : = (1/Z_T) \exp (H_T) dP$$, where $$H_T$$ imparts the self-attracting interaction of the paths up to $$T$$, and $$Z_T$$ is the appropriate normalizing constant. The purpose of the paper is a discussion of the behavior of $$P_T$$ as $$T \to \infty$$ for specific $$H_T$$. The authors consider the cases of $$H_T$$ given by a potential function $$V$$ on $$Z^d$$ as well as $$H_T = - N_T$$, where $$N_T$$ denotes the number of points visited by the random walk up to time $$T$$. In both situations the typical paths under $$P_T$$ as $$T \to \infty$$ clump together much more than those of the free random walk and give rise to localization phenomena. The paper mainly discusses the results contained in [first author, Ann. Probab. 22, No. 2, 875-918 (1994; Zbl 0819.60028) and the authors, “On self-attracting $$d$$- dimensional random waves” (Preprint, 1994)].
For the entire collection see [Zbl 0814.00017].
Reviewer: O.Brockhaus (Bonn)

### MSC:

 60F10 Large deviations 60F05 Central limit and other weak theorems 60K35 Interacting random processes; statistical mechanics type models; percolation theory

Zbl 0819.60028