  The self-repelling random walk with directed edges was introduced by T\'oth
and Vet\H{o} 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. T\'oth and Vet\H{o}
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.
