Summary
Consider a one-dimensional walk (S k ) k having steps of bounded size, and weight the probability of the path with some factor 1−α∈(0,1) for every single self-intersection up to timen. We prove thatS n /S S converges towards some deterministic number called the effective drift of the self-repellent walk. Furthermore, this drift is shown to tend to the basic drift as α tends to 0 and, as α tends to 1, to the self-avoiding walk's drift which is introduced in [10]. The main tool of the present paper is a representation of the sequence of the local times as a functional of a certain Markov process.
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Partially supported by Swiss National Sciences Foundation Grant 20-36305.92
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König, W. The drift of a one-dimensional self-repellent random walk with bounded increments. Probab. Th. Rel. Fields 100, 513–544 (1994). https://doi.org/10.1007/BF01268992
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DOI: https://doi.org/10.1007/BF01268992
Mathematics Subject Classfication (1991)
- 60K35
- 58E30
- 60F10
- 60J15