Cutoff for random walks on dihedral groups
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915571348537344 |
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| author | Huang, Xiangying Rao, Renyu |
| author_facet | Huang, Xiangying Rao, Renyu |
| contents | We study the random walk on a finite dihedral group $G$ driven by the uniform measure on $k$ independently and uniformly chosen elements. We show that the walk exhibits cutoff with high probability throughout nearly the entire regime $1 \ll \log k \ll \log |G|$, and determine the precise cutoff time. Interestingly, this mixing time differs from the entropic time that characterizes cutoff behavior for random walks on Abelian groups. When $k \gg \log|G|$ and $\log k \ll \log|G|$, cutoff occurs with high probability on random Cayley graphs of virtually Abelian groups. The analysis develops techniques for obtaining sharper entropic estimates of an auxiliary process on high-dimensional lattices with dependent coordinates, which may also prove useful for related models in broader contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cutoff for random walks on dihedral groups Huang, Xiangying Rao, Renyu Probability 05C81, 60B15, 60J27 We study the random walk on a finite dihedral group $G$ driven by the uniform measure on $k$ independently and uniformly chosen elements. We show that the walk exhibits cutoff with high probability throughout nearly the entire regime $1 \ll \log k \ll \log |G|$, and determine the precise cutoff time. Interestingly, this mixing time differs from the entropic time that characterizes cutoff behavior for random walks on Abelian groups. When $k \gg \log|G|$ and $\log k \ll \log|G|$, cutoff occurs with high probability on random Cayley graphs of virtually Abelian groups. The analysis develops techniques for obtaining sharper entropic estimates of an auxiliary process on high-dimensional lattices with dependent coordinates, which may also prove useful for related models in broader contexts. |
| title | Cutoff for random walks on dihedral groups |
| topic | Probability 05C81, 60B15, 60J27 |
| url | https://arxiv.org/abs/2510.19942 |