Elephant random walk on the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$
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| Format: | Preprint |
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2026
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| _version_ | 1866908940169641984 |
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| author | Mukherjee, Soumendu Sundar Talukdar, Himasish |
| author_facet | Mukherjee, Soumendu Sundar Talukdar, Himasish |
| contents | 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 $\mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04922 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Elephant random walk on the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$ Mukherjee, Soumendu Sundar Talukdar, Himasish Probability Statistical Mechanics Mathematical Physics 60G50, 82C41, 60K99, 60G42 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 $\mathbb{Z}$. |
| title | Elephant random walk on the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$ |
| topic | Probability Statistical Mechanics Mathematical Physics 60G50, 82C41, 60K99, 60G42 |
| url | https://arxiv.org/abs/2604.04922 |