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

TL;DR

This paper investigates the fluctuations of local times in a non-Markovian self-repelling random walk with directed edges, revealing convergence properties of these fluctuations around a deterministic limit.

Contribution

It provides the first analysis of the fluctuation behavior of local times in this model, showing convergence in specific topologies and extending understanding of its stochastic properties.

Findings

Fluctuations converge in Skorohod M1 topology.

Convergence also holds in uniform topology away from discontinuities.

Fluctuations of the stopping times also converge.

Abstract

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, as 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 deterministic process, instead of a random one as in similar models. In this work, we study the fluctuations of the local times process around its 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.