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)


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


Zbl 0819.60028