Cutoff phenomenon for the warp-transpose top with random shuffle
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866912329954754560 |
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| author | Ghosh, Subhajit |
| author_facet | Ghosh, Subhajit |
| contents | Let $\{G_n\}_1^{\infty}$ be a sequence of non-trivial finite groups. In this paper, we study the properties of a random walk on the complete monomial group $G_n\wr S_n$ generated by the elements of the form $(\text{e},\dots,\text{e},g;\text{id})$ and $(\text{e},\dots,\text{e},g^{-1},\text{e},\dots,\text{e},g;(i,n))$ for $g\in G_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is $O\left(n\log n+\frac{1}{2}n\log (|G_n|-1)\right)$. We show that this shuffle exhibits $\ell^2$-cutoff at $n\log n+\frac{1}{2}n\log (|G_n|-1)$ and total variation cutoff at $n\log n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_00533 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cutoff phenomenon for the warp-transpose top with random shuffle Ghosh, Subhajit Probability 60J10, 60B15, 60C05 Let $\{G_n\}_1^{\infty}$ be a sequence of non-trivial finite groups. In this paper, we study the properties of a random walk on the complete monomial group $G_n\wr S_n$ generated by the elements of the form $(\text{e},\dots,\text{e},g;\text{id})$ and $(\text{e},\dots,\text{e},g^{-1},\text{e},\dots,\text{e},g;(i,n))$ for $g\in G_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is $O\left(n\log n+\frac{1}{2}n\log (|G_n|-1)\right)$. We show that this shuffle exhibits $\ell^2$-cutoff at $n\log n+\frac{1}{2}n\log (|G_n|-1)$ and total variation cutoff at $n\log n$. |
| title | Cutoff phenomenon for the warp-transpose top with random shuffle |
| topic | Probability 60J10, 60B15, 60C05 |
| url | https://arxiv.org/abs/2101.00533 |