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

Bolthausen, Erwin; Schmock, Uwe

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

Keywords:

nearest-neighbor random walk; self-attracting interaction; localization phenomena

Citations:

Zbl 0819.60028