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Recurrence of the plane Elephant random walk

Nicolas Curien, Lucile Laulin

TL;DR

This work proves recurrence for the planar elephant random walk in the diffusive regime $\alpha<\frac{1}{2}$ by a concise comparison to the standard planar random walk. It exploits a Radon–Nikodym contiguity framework between the elephant and simple random walks, together with a four-color urn structure for the directional counts and a diffusion-limit result for $D^{X}_n(\mathbf{e}_i)/\sqrt{n}$. The authors derive a non-quantitative but robust lower bound on return probabilities and, via Jeulin’s lemma and a second-moment argument, establish almost-sure recurrence in 2D. The method complements previous work by Qin (2023) and offers a simpler approach that may extend to other reinforced random-walk settings, while not covering the critical case $\alpha=\frac{1}{2}$ or higher dimensions where transience occurs.

Abstract

We give a short proof of the recurrence of the two-dimensional elephant random walk in the diffusive regime. This was recently established by Shuo Qin, but our proof only uses very rough comparison with the standard plane random walk. We hope that the method can be useful for other applications.

Recurrence of the plane Elephant random walk

TL;DR

This work proves recurrence for the planar elephant random walk in the diffusive regime by a concise comparison to the standard planar random walk. It exploits a Radon–Nikodym contiguity framework between the elephant and simple random walks, together with a four-color urn structure for the directional counts and a diffusion-limit result for . The authors derive a non-quantitative but robust lower bound on return probabilities and, via Jeulin’s lemma and a second-moment argument, establish almost-sure recurrence in 2D. The method complements previous work by Qin (2023) and offers a simpler approach that may extend to other reinforced random-walk settings, while not covering the critical case or higher dimensions where transience occurs.

Abstract

We give a short proof of the recurrence of the two-dimensional elephant random walk in the diffusive regime. This was recently established by Shuo Qin, but our proof only uses very rough comparison with the standard plane random walk. We hope that the method can be useful for other applications.
Paper Structure (4 sections, 4 theorems, 20 equations, 1 figure)

This paper contains 4 sections, 4 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation.
  3. Comparison between elephant and simple random walk
  4. Proof of Theorem \ref{['theo:rec']}

Key Result

Theorem 1.1

In the diffusive regime $\alpha < \alpha_c = \frac{1}{2}$, the plane elephant random walk is recurrent.

Figures (1)

  • Figure 2.1: Illustration of the proof of Proposition \ref{['prop:retour']}. Conditionally on $\mathcal{F}_n$ and on the fact that the counting directions processes are controlled at time $n$, the blue and red parts are independent on events of large probability. This is sufficient to imply a lower bound on the probability of return to $\mathbf{0}$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Proposition 2.1: Markov contiguity
  • Proposition 2.2
  • Lemma 2.3