Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Once-reinforced random walk in high dimensions

Dor Elboim, Gady Kozma

TL;DR

This work analyzes the once-reinforced random walk (ORRW) on $\mathbb{Z}^d$ for $d\ge6$, proving transience when the reinforcement is small and showing a diffusive scaling limit via a coupling to Brownian motion. The authors develop a robust multiscale induction built on many relaxed times, a capacity-based control of the walk's history, and a demon mechanism that induces approximate independence between spatial blocks. The key technical advances are a capacity-to-volume estimate in high dimensions, a demon-driven supermartingale framework, and a concatenation/KMT-type coupling that yields a central limit-type behavior. The results illuminate high-dimensional self-interacting walks and establish a flexible toolkit potentially applicable to other self-interacting models and related diffusion limits. Altogether, the paper makes rigorous progress on Sidoravicius' conjecture by confirming the transient, diffusive regime in $d\ge6$ for small reinforcement and linking the dynamics to Brownian motion.

Abstract

We study the once-reinforced random walk on $\mathbb Z^d$, which is a self-interacting walk that has a higher probability to cross edges that were already visited. We prove that the walk is transient when $d\ge 6$ and when the reinforcement is small, establishing a conjecture of Sidoravicius in these dimensions. Moreover, in this case we prove that the walk behaves diffusively and can be coupled with Brownian motion. One of the main ideas in the proof is a certain capacity estimate which shows that the trajectory of the walk is nowhere heavy. We also use a game-theoretic-type ingredient that we call ``the demon" to force spatial independence in the process.

Once-reinforced random walk in high dimensions

TL;DR

This work analyzes the once-reinforced random walk (ORRW) on for , proving transience when the reinforcement is small and showing a diffusive scaling limit via a coupling to Brownian motion. The authors develop a robust multiscale induction built on many relaxed times, a capacity-based control of the walk's history, and a demon mechanism that induces approximate independence between spatial blocks. The key technical advances are a capacity-to-volume estimate in high dimensions, a demon-driven supermartingale framework, and a concatenation/KMT-type coupling that yields a central limit-type behavior. The results illuminate high-dimensional self-interacting walks and establish a flexible toolkit potentially applicable to other self-interacting models and related diffusion limits. Altogether, the paper makes rigorous progress on Sidoravicius' conjecture by confirming the transient, diffusive regime in for small reinforcement and linking the dynamics to Brownian motion.

Abstract

We study the once-reinforced random walk on , which is a self-interacting walk that has a higher probability to cross edges that were already visited. We prove that the walk is transient when and when the reinforcement is small, establishing a conjecture of Sidoravicius in these dimensions. Moreover, in this case we prove that the walk behaves diffusively and can be coupled with Brownian motion. One of the main ideas in the proof is a certain capacity estimate which shows that the trajectory of the walk is nowhere heavy. We also use a game-theoretic-type ingredient that we call ``the demon" to force spatial independence in the process.
Paper Structure (29 sections, 37 theorems, 147 equations, 1 figure)

This paper contains 29 sections, 37 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sidoravicius' conjecture
  3. Previous work
  4. The method of proof
  5. Comparison with Elboim and Sly, 2022
  6. Acknowledgments
  7. Preliminaries
  8. Setting up the induction
  9. Demons
  10. Heavy blocks and relaxed times
  11. The inductive assumptions
  12. The induction base and step
  13. Concatenation
  14. Avoiding the history
  15. Concatenation
  16. ...and 14 more sections

Key Result

Theorem 1

Suppose that $d\ge 6$ and let $a>0$ sufficiently small. Then once-reinforced walk $W$ on $\mathbb Z ^d$ with reinforcement $a$ is transient. Moreover, there is a constant $\sigma \in [\frac{0.9}{\sqrt{d}},\frac{1.1}{\sqrt{d}}]$ depending on $a$ and $d$ such that where $\{B(s)\}_{s\ge 0}$ is a standard Brownian motion in $\mathbb R^d$. Here the convergence is weak convergence in $C[0,T]$ for all $

Figures (1)

  • Figure 1: The three scales in the proof of Lemma \ref{['lem:Cap-Vol']}. The left pane shows the large scale ($R$): the walk $W_x$ starting from $x\in A\cap \Lambda$, exits $A\cup \Lambda$ quickly, doesn't get close again, enters the inner box $[-R,R]^d$, and spends $R^2$ time in there. The middle pane, at scale $r$, shows the last exit of the ORRW from $A$ and the path $W_z$ in blue. The right pane, at scale $r^{7\epsilon}$, shows the coupling in more detail: The red path is $W_x$ (and $W_1+x$, which is its section until $y$). The blue path is $W_z$ (and $W_1'+z$, which is its first section). The green paths are the two translations of $W_2$.

Theorems & Definitions (77)

  • Theorem 1
  • Definition 2.1: Heavy blocks
  • Definition 2.2: Relaxed times
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2
  • Lemma 3.1
  • proof
  • Lemma 3.2: Virgin relaxed times
  • ...and 67 more