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

Laure Marêché

TL;DR

This work analyzes fluctuations of the local times for the self-repelling random walk with directed edges on $\mathbb{Z}$, revealing that centered fluctuations converge to a Brownian-type limit restricted to the deterministic support, while the classical Skorohod $J_1$ topology is too strong to capture this limit. The authors develop a Ray-Knight-type coupling to approximate local-time increments by i.i.d. variables, establish a precise $\,M_1$-topology framework, and construct parametric representations to compare the fluctuations with a renormalized sum of i.i.d. variables. They show the limiting fluctuation process is $\big(B_y^x \mathds{1}_{\{y\in[-|x|-2\theta,|x|+2\theta)\}}\big)_{y}$, where $B^x$ is a Brownian motion with variance $\mathrm{Var}(\rho_-)$, and prove that the associated stopping-time fluctuations are Gaussian with a computable variance. In addition to establishing the $M_1$-limit and the non-$J_1$ convergence, the paper provides a uniform-convergence result away from the limit's discontinuities, yielding a comprehensive picture of fluctuation phenomena around the deterministic triangle limit. The findings highlight how non-Markovian, self-interacting walks can exhibit diffusive fluctuations around a deterministic backbone, with topology playing a critical role in identifying the correct mode of convergence for discontinuous limits.

Abstract

In 2008, Tóth and Vető 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.

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

TL;DR

This work analyzes fluctuations of the local times for the self-repelling random walk with directed edges on , revealing that centered fluctuations converge to a Brownian-type limit restricted to the deterministic support, while the classical Skorohod topology is too strong to capture this limit. The authors develop a Ray-Knight-type coupling to approximate local-time increments by i.i.d. variables, establish a precise -topology framework, and construct parametric representations to compare the fluctuations with a renormalized sum of i.i.d. variables. They show the limiting fluctuation process is , where is a Brownian motion with variance , and prove that the associated stopping-time fluctuations are Gaussian with a computable variance. In addition to establishing the -limit and the non- convergence, the paper provides a uniform-convergence result away from the limit's discontinuities, yielding a comprehensive picture of fluctuation phenomena around the deterministic triangle limit. The findings highlight how non-Markovian, self-interacting walks can exhibit diffusive fluctuations around a deterministic backbone, with topology playing a critical role in identifying the correct mode of convergence for discontinuous limits.

Abstract

In 2008, Tóth and Vető defined the self-repelling random walk with directed edges as a non-Markovian random walk on : in this model, the probability that the walk moves from a point of 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 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.
Paper Structure (19 sections, 31 theorems, 56 equations, 2 figures)

This paper contains 19 sections, 31 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction and results
  2. Self-interacting random walks
  3. The self-repelling random walk with directed edges
  4. Topologies for the convergence of the local times process
  5. Results
  6. Proof ideas
  7. Organization of the paper
  8. Coupling of the local times increments with i.i.d. random variables
  9. Where the local times approach 0
  10. Place where we hit 0
  11. Control of low local times
  12. Skorohod $M_1$ distance
  13. Case $\mathbf{a\in(0,|x|+2\theta-N^{-1/8})}$
  14. Case $\mathbf{a>|x|+2\theta+N^{-1/8}}$
  15. Convergence of the local times process: proof of Theorem \ref{['thm_main']} and Proposition \ref{['prop_uniform_conv']}
  16. ...and 4 more sections

Key Result

Theorem 1

For any $\theta>0$, $x\in \mathds{R}$, then $\sup_{y \in \mathds{R}}|\frac{1}{N}\ell^+(T_{\lfloor N \theta\rfloor,\lfloor N x\rfloor}^\pm,\lfloor N y\rfloor)-(\frac{|x|-|y|}{2}+\theta)_+|$ converges in probability to 0 when $N$ tends to $+\infty$.

Figures (2)

  • Figure 1: The successive steps of the parametric representations of $Y_N^-|_{[-a,a]}$ and $Y_N|_{[-a,a]}$ if $I^+ \leq \lfloor(|x|+2\theta)N\rfloor$. At each step, the parts of the graphs the parametric representations travel through are thickened.
  • Figure 2: The successive steps of the parametric representations of $Y_N^-|_{[-a,a]}$ and $Y_N|_{[-a,a]}$ if $I^+ > \lfloor(|x|+2\theta)N\rfloor$. At each step, the parts of the graphs the parametric representations travel through are thickened.

Theorems & Definitions (61)

  • Theorem : Theorem 1 of Toth_et_al2008
  • Proposition : Corollary 1 of Toth_et_al2008
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 5
  • Remark 6
  • Lemma 7: Lemma 1 of Toth_et_al2008
  • Lemma 8
  • ...and 51 more