Cutoff in total variation for the shelf shuffle
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916449012940800 |
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| author | Ottolini, Andrea Chen, Ray |
| author_facet | Ottolini, Andrea Chen, Ray |
| contents | We analyze the mixing time of a popular shuffling machine known as the shelf shuffler. It is a modified version of a $2m$-handed riffle shuffle ($m=10$ in casinos) in which a deck of $n$ cards is split multinomially into $2m$ piles, the even-numbered piles are reversed, and then cards are dropped from piles proportionally to their sizes. We prove that $\frac{5}{4} \log_{2m} n$ shuffles are necessary and sufficient to mix in total variation, and a cutoff occurs with constant window size. We also determine the cutoff profile in terms of the total variation distance between two shifted normal random variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cutoff in total variation for the shelf shuffle Ottolini, Andrea Chen, Ray Probability 60J10 We analyze the mixing time of a popular shuffling machine known as the shelf shuffler. It is a modified version of a $2m$-handed riffle shuffle ($m=10$ in casinos) in which a deck of $n$ cards is split multinomially into $2m$ piles, the even-numbered piles are reversed, and then cards are dropped from piles proportionally to their sizes. We prove that $\frac{5}{4} \log_{2m} n$ shuffles are necessary and sufficient to mix in total variation, and a cutoff occurs with constant window size. We also determine the cutoff profile in terms of the total variation distance between two shifted normal random variables. |
| title | Cutoff in total variation for the shelf shuffle |
| topic | Probability 60J10 |
| url | https://arxiv.org/abs/2410.17345 |