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Elephant random walk on the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$

Soumendu Sundar Mukherjee, Himasish Talukdar

Abstract

Elephant random walks were studied recently in \cite{mukherjee2025elephant} on the groups $\mathbb{Z}^{*d_1} * \mathbb{Z}_2^{*d_2}$ whose Cayley graphs are infinite $d$-regular trees with $d = 2d_1 + d_2$. It was found that for $d \ge 3$, the elephant walk is ballistic with the same asymptotic speed $\frac{d - 2}{d}$ as the simple random walk and the memory parameter appears only in the rate of convergence to the limiting speed. In the $d = 2$ case, there are two such groups, both having the bi-infinite path as their Cayley graph. For $(d_1, d_2) = (1, 0)$, the walk is the usual elephant random walk on $\mathbb{Z}$, which exhibits anomalous diffusion. In this article, we study the other case, namely $(d_1, d_2) = (0, 2)$, which corresponds to the infinite dihedral group $D_\infty \cong \mathbb{Z}_2 * \mathbb{Z}_2$. Unlike the classical ERW on $\mathbb{Z}$, which is a time-inhomogeneous Markov chain, the ERW on $D_{\infty}$ is non-Markovian. We show that the first and second order behaviours of the \emph{signed location} of the walker agree with those of the simple symmetric random walk on $\mathbb{Z}$, with the memory parameter essentially manifesting itself via a lower order correction term that can be written as an explicit functional of the elephant walk on $\mathbb{Z}$. Our result demonstrates that unlike the simple random walk, the elephant walk is sensitive to local algebraic relations. Indeed, although $D_{\infty}$ is virtually abelian, containing $\mathbb{Z}$ as a finite-index subgroup, the involutive nature of its generators effectively neutralises memory, thereby ruling out any potential superdiffusive behaviour, in contrast to the superdiffusion observed on its abelian cousin $\mathbb{Z}$.

Elephant random walk on the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$

Abstract

Elephant random walks were studied recently in \cite{mukherjee2025elephant} on the groups whose Cayley graphs are infinite -regular trees with . It was found that for , the elephant walk is ballistic with the same asymptotic speed as the simple random walk and the memory parameter appears only in the rate of convergence to the limiting speed. In the case, there are two such groups, both having the bi-infinite path as their Cayley graph. For , the walk is the usual elephant random walk on , which exhibits anomalous diffusion. In this article, we study the other case, namely , which corresponds to the infinite dihedral group . Unlike the classical ERW on , which is a time-inhomogeneous Markov chain, the ERW on is non-Markovian. We show that the first and second order behaviours of the \emph{signed location} of the walker agree with those of the simple symmetric random walk on , with the memory parameter essentially manifesting itself via a lower order correction term that can be written as an explicit functional of the elephant walk on . Our result demonstrates that unlike the simple random walk, the elephant walk is sensitive to local algebraic relations. Indeed, although is virtually abelian, containing as a finite-index subgroup, the involutive nature of its generators effectively neutralises memory, thereby ruling out any potential superdiffusive behaviour, in contrast to the superdiffusion observed on its abelian cousin .

Paper Structure

This paper contains 4 sections, 11 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model and our main results
  3. Proofs
  4. Auxiliary results

Key Result

Theorem 2.1

There exists a zero-mean martingale $(\Xi_n)_{n \ge 0}$ adapted to $(\mathcal{F}_{n})_{n \ge 0}$, with bounded increments and its predictable quadratic variation $\langle \Xi \rangle_n$ satisfying $\frac{\langle \Xi \rangle_n}{n} \xrightarrow{\text{a.s.}} 1$, such that where $q = 2p - 1 \in [-1, 1)$. The predictable process $(Z_n)_{n \ge 0}$, where $Z_n := \Delta_{n}^{(s)} - \Xi_n$, converges alm

Figures (2)

  • Figure 1: The (left-) Cayley graph of $D_{\infty}$ with respect to the generators $\{a, b\}$. The integer below a node denote its signed location.
  • Figure 2: $\mathbb{E}[Z_{\infty}^2]$ as a function of $q = 2p - 1$. The plot illustrates the slow rate of convergence for $q$ close to $1$.

Theorems & Definitions (24)

  • Remark 2.1
  • Theorem 2.1
  • Corollary 2.1
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof : Proof of Corollary \ref{['cor:S_n']}
  • ...and 14 more