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 as a nearest-neighbor random walk on $\mathbb{Z}$ 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 $(X_m)_{m\in\mathbb{N}}$, for any $t \geq 0$ the quantity $\frac{1}{\sqrt{N}}X_{\lfloor Nt \rfloor}$ converges in distribution to a non-trivial limit when $N$ tends to $+\infty$, but the process $(\frac{1}{\sqrt{N}}X_{\lfloor Nt \rfloor})_{t \geq 0}$ does not converge in distribution. In this paper, we prove not only that $(\frac{1}{\sqrt{N}}X_{\lfloor Nt \rfloor})_{t \geq 0}$ 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 $T_N$ in a large family of stopping times, the process $(\frac{1}{n}(X_{T_N+tn^{3/2}}-X_{T_N}))_{t \geq 0}$ 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.