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Recurrence and transience of multidimensional elephant random walks

Shuo Qin

TL;DR

The paper resolves fundamental questions about recurrence and transience for the multidimensional elephant random walk (MERW) across dimensions, proving transience for $d\ge 3$ and identifying precise phase transitions in low dimensions at $p=3/4$ ($d=1$) and $p=5/8$ ($d=2$). It develops a unified framework built on coupling with one-dimensional elephant random walks, a $d$-ERW construction, and a Polya urn continuous-time embedding, complemented by Lyapunov-function methods to handle the non-Markovian setting. The results include sharp rate estimates for escape, finite expected numbers of zeros in high dimensions, detailed limiting behavior in the superdiffusive regime via the variables $Y_d$ and $W_d$, and existence of densities for the limiting distributions in low dimensions. The study advances the understanding of memory-driven random walks and offers new tools (coupling, urn embedding, Lyapunov methods) applicable to reinforced stochastic processes and non-Markovian dynamics.

Abstract

We prove a conjecture by Bertoin that the multi-dimensional elephant random walk on $\mathbb{Z}^d$($d\geq 3$) is transient and the expected number of zeros is finite. We also provide some estimates on the rate of escape. In dimensions $d= 1, 2$, we prove that phase transitions between recurrence and transience occur at $p=(2d+1)/(4d)$. Let $S$ be an elephant random walk with parameter $p$. For $p \leq 3/4$, we provide a Berry-Esseen type bound for properly normalized $S_n$. For $p>3/4$, the distribution of $\lim_{n\to \infty} S_n/n^{2p-1}$ will be studied.

Recurrence and transience of multidimensional elephant random walks

TL;DR

The paper resolves fundamental questions about recurrence and transience for the multidimensional elephant random walk (MERW) across dimensions, proving transience for and identifying precise phase transitions in low dimensions at () and (). It develops a unified framework built on coupling with one-dimensional elephant random walks, a -ERW construction, and a Polya urn continuous-time embedding, complemented by Lyapunov-function methods to handle the non-Markovian setting. The results include sharp rate estimates for escape, finite expected numbers of zeros in high dimensions, detailed limiting behavior in the superdiffusive regime via the variables and , and existence of densities for the limiting distributions in low dimensions. The study advances the understanding of memory-driven random walks and offers new tools (coupling, urn embedding, Lyapunov methods) applicable to reinforced stochastic processes and non-Markovian dynamics.

Abstract

We prove a conjecture by Bertoin that the multi-dimensional elephant random walk on () is transient and the expected number of zeros is finite. We also provide some estimates on the rate of escape. In dimensions , we prove that phase transitions between recurrence and transience occur at . Let be an elephant random walk with parameter . For , we provide a Berry-Esseen type bound for properly normalized . For , the distribution of will be studied.
Paper Structure (24 sections, 25 theorems, 196 equations, 1 figure)

This paper contains 24 sections, 25 theorems, 196 equations, 1 figure.

Table of Contents

  1. General introduction
  2. Definitions and main results
  3. Introduction to the proofs and the techniques
  4. Notation
  5. Coupling
  6. Connection to Pólya-type urns and continuous-time embedding
  7. Lyapunov functions method
  8. Organization of the paper
  9. Proofs of the main results
  10. Coupling
  11. Coupling of ERWs: Proof of Proposition \ref{['monotonicitythm']}
  12. Coupling of a MERW and d-ERWs
  13. Proof of Theorem \ref{['MERWtran12d']}
  14. The limit in the superdiffusive regime
  15. Proof of Proposition \ref{['Lnot0MERW']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $S=(S_n)_{n\in \mathbb{N}}$ be a MERW on $\mathbb{Z}^d$ with $d\geq 3$ and memory parameter $p$. Then, for any $p \in [0,1]$, $S$ is transient. More precisely, for any $\nu \in (0,\frac{1}{2}-\frac{1}{d})$, in particular, almost surely, $\|S_n\|> n^{\nu}$ for large $n$.

Figures (1)

  • Figure 1: Simulation results of 4 independent MERWs on $\mathbb{Z}^2$ with $n=1000000$ steps, showing the range of the walks

Theorems & Definitions (61)

  • Remark 1.1
  • Definition 1
  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.2
  • Theorem 1.4
  • Remark 1.3
  • Proposition 1.5
  • Remark 1.4
  • ...and 51 more