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

Abstract

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 ${(X_m)}_{m\in \mathbb{N}}$, for any $t\ge 0$ the quantity $\frac{1}{\sqrt{N}}{X}_{\lfloor Nt\rfloor }$ converges in distribution to a non-trivial limit when N tends to $+\mathrm{\infty }$, but the process ${(\frac{1}{\sqrt{N}}{X}_{\lfloor Nt\rfloor })}_{t\ge 0}$ does not converge in distribution. In this paper, we prove not only that ${(\frac{1}{\sqrt{N}}{X}_{\lfloor Nt\rfloor })}_{t\ge 0}$ 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 ${(\frac{1}{n}(X_{T_N+t{n}^{3∕2}}-X_{T_N}))}_{t\ge 0}$ 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.