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 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 $t\ge 0$ 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 $(\frac{1}{\sqrt{N}}{X}_{\lfloor Nt\rfloor }{)}_{t\ge 0}$ 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.

65 pages, no figure

References in corpus (3)

• Self-repelling random walk with directed edges on Z
• Central Limit Theorem for the self-repelling random walk with directed edges
• Fluctuations of the local times of the self-repelling random walk with directed edges

Cited by in corpus (2)

• Fluctuations of the local times of the self-repelling random walk with directed edges
• Velocity trapping in the lifted TASEP and the true self-avoiding random walk