The drift of a one-dimensional self-repellent random walk with bounded increments

Abstract

Consider a one-dimensional walk (Sk)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 time n. We prove that Sn/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 α tends to 0 and, as α tends to 1, to the self-avoiding walk's drift which was introduced in an earlier paper of ours [Probab. Theory Related Fields 96 (1993), no. 4, 521--543]. 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.