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On self-attracting random walks


Bolthausen, E; Schmock, U (1995). On self-attracting random walks. In: Cranston, M C; Pinsky, M A. Stochastic analysis (Ithaca, NY, 1993). Providence, RI: American Mathematical Society, 23-44.

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.

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.

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Additional indexing

Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:nearest-neighbor random walk; self-attracting interaction; localization phenomena
Language:English
Date:1995
Deposited On:20 May 2010 12:53
Last Modified:05 Apr 2016 13:27
Publisher:American Mathematical Society
Series Name:Proceedings of Symposia in Pure Mathematics
Number:57
ISSN:0082-0717
ISBN:0-8218-0289-5
Additional Information:First published in [On self-attracting random walks. Stochastic analysis (Ithaca, NY, 1993), 23--44, Proc. Sympos. Pure Math., 57], published by the American Mathematical Society
Official URL:http://www.ams.org/bookstore?fn=20&arg1=pspumseries&ikey=PSPUM-57
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0829.60021
http://www.fam.tuwien.ac.at/~schmock/Attracting_walks.html
Permanent URL: http://doi.org/10.5167/uzh-22578

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