Limit theorems for the trajectory of the self-repelling random walk with directed edges

The self-repelling random walk with directed edges was introduced by Tóth and Vető in 2008 [23] as a nearest-neighbor random walk on Z that is non-Markovian: at each step, the probability to cross a directed edge depends on the number of previous crossings of this directed edge. Tóth and Vető found this walk to have a very peculiar behavior, and conjectured that, denoting the walk by (Formula presented), for any t ≥ 0 the quantity (Formula presented) converges in distribution to a non-trivial limit when N tends to +∞, but the process (Formula presented) does not converge in distribution. In this paper, we prove not only that (Formula presented) admits no limit in distribution in the standard Skorohod topology, but more importantly that the trajectories of the random walk still satisfy another limit theorem, of a new kind. Indeed, we show that for n suitably smaller than N and TN in a large family of stopping times, the process (Formula presented) admits a non-trivial limit in distribution. The proof partly relies on combinations of reflected and absorbed Brownian motions which may be interesting in their own right.