The drift of a one-dimensional self-repellent random walk with bounded increments. (English) Zbl 0810.60095

Summary: Consider a one-dimensional walk \((S_ k)_ k\) having steps of bounded size, and weight the probability of the path with some factor \(1 - \alpha \in (0,1)\) for every single self-intersection up to time \(n\). We prove that \(S_ n/n\) 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 \(\alpha\) tends to 0 and, as \(\alpha\) tends to 1, to the self-avoiding walk’s drift which is introduced in [author, ibid. 96, No. 4, 521-543 (1993; Zbl 0792.60097)]. 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.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
58E30 Variational principles in infinite-dimensional spaces
60F10 Large deviations
60G50 Sums of independent random variables; random walks


Zbl 0792.60097
Full Text: DOI


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