The $k$-cycle shuffling with repeated cards
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910082253455360 |
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| author | Shen, Jiahe |
| author_facet | Shen, Jiahe |
| contents | We investigate the $k$-cycle shuffle on repeated cards, namely on a deck consisting of $l$ identical copies of each of $m$ card types, with total size $n=ml$. We establish asymptotic results for the total variation mixing of this shuffle, including cutoff and explicit limiting profiles. For fixed $l$, we show that the walk exhibits cutoff at time $\frac{n}{k}\log n$ with window of order $\frac{n}{k}$, and we identify the limiting profile in terms of the total variation distance between Poisson distributions arising from quotient fixed-point statistics. When $l\to\infty$ with sufficiently slow growth, more precisely when $l=o(\log n)$, we prove that the cutoff location shifts to $\frac{n}{k}\left(\log n-\frac 12\log l\right)$, again with window of order $\frac{n}{k}$, and that the limiting profile is asymptotically Gaussian, arising from a Poisson comparison after normal approximation.
The proof is based on an approximation of the shuffling measure by an explicitly tractable auxiliary measure, generalizing the $k=2$ case from Jain and Sawhney (arXiv:2410.23944). The representation-theoretic framework underlying the analysis of this auxiliary measure follows from the work of Hough (arXiv:1605.00911) and Nestoridi and Olesker-Taylor (arXiv:2005.13437) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27433 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $k$-cycle shuffling with repeated cards Shen, Jiahe Probability Combinatorics Representation Theory We investigate the $k$-cycle shuffle on repeated cards, namely on a deck consisting of $l$ identical copies of each of $m$ card types, with total size $n=ml$. We establish asymptotic results for the total variation mixing of this shuffle, including cutoff and explicit limiting profiles. For fixed $l$, we show that the walk exhibits cutoff at time $\frac{n}{k}\log n$ with window of order $\frac{n}{k}$, and we identify the limiting profile in terms of the total variation distance between Poisson distributions arising from quotient fixed-point statistics. When $l\to\infty$ with sufficiently slow growth, more precisely when $l=o(\log n)$, we prove that the cutoff location shifts to $\frac{n}{k}\left(\log n-\frac 12\log l\right)$, again with window of order $\frac{n}{k}$, and that the limiting profile is asymptotically Gaussian, arising from a Poisson comparison after normal approximation. The proof is based on an approximation of the shuffling measure by an explicitly tractable auxiliary measure, generalizing the $k=2$ case from Jain and Sawhney (arXiv:2410.23944). The representation-theoretic framework underlying the analysis of this auxiliary measure follows from the work of Hough (arXiv:1605.00911) and Nestoridi and Olesker-Taylor (arXiv:2005.13437) |
| title | The $k$-cycle shuffling with repeated cards |
| topic | Probability Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2603.27433 |