Fluctuations of the local times of the self-repelling random walk with directed edges

In 2008, T\'oth and Vet\H{o} defined the self-repelling random walk with directed edges as a non-Markovian random walk on $\mathbb{Z}$: in this model, the probability that the walk moves from a point of $\mathbb{Z}$ to a given neighbor depends on the number of previous crossings of the directed edge from the initial point to the target, called the local time of the edge. They found this model had a very peculiar behavior. Indeed, for the non-Markovian random walks most closely related to it, which are defined with undirected edges replacing directed edges, T\'oth proved in his works of 1994, 1995 and 1996 that the process formed by the local times of all the edges, evaluated at a stopping time of a certain type and suitably renormalized, converges to a random process. The behavior of the self-repelling random walk with directed edges is very different and unusual, as T\'oth and Vet\H{o} proved in 2008 that the limit process is deterministic. In this work, we study the fluctuations of the local times process around this deterministic limit, about which nothing was previously known. We prove that these fluctuations converge in the Skorohod $M_1$ topology, as well as in the uniform topology away from the discontinuities of the limit, but not in the most classical Skorohod topology. We also prove the convergence of the fluctuations of the aforementioned stopping times.Comment: 31 pages, 2 figure