Explicit cutoff profiles for colored top-$m$-to-random shuffles
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arXiv
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| Format: | Preprint |
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2026
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| author | Feng, Ivan Z. |
| author_facet | Feng, Ivan Z. |
| contents | We study $p$-colored top-$m$-to-random on the wreath product $G_{n,p}=C_p\wr S_n$, with $m$ fixed. Using the Nakano-Sadahiro-Sakurai basis elements $B_m$, we obtain exact nested-set occupancy mixtures and reduce the likelihood ratio to the single statistic $L_p$. This yields exact formulas for separation and $L^\infty(U)$, and exact one-dimensional formulas for total variation, $L^q(U)$ ($1\le q<\infty$), $χ^2$, and relative entropy. At $k=\Bigl\lfloor \frac{n}{m}(\log n+c)\Bigr\rfloor$, the number of never-chosen labels in the associated $m$-subset occupancy model converges in law to $\mathrm{Poisson}(e^{-c})$, giving the total-variation profile $f_p(c)$, the separation profile, and the corresponding $L^q(U)$, $L^\infty(U)$, $χ^2$, and relative-entropy profiles. For $m=1$ we recover colored top-to-random; for $p=1$, the total-variation profile reduces to the Diaconis-Fill-Pitman profile. For the reversed chain, we also identify optimal strong stationary times whose tail probabilities are exactly the separation distances. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_09933 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit cutoff profiles for colored top-$m$-to-random shuffles Feng, Ivan Z. Probability Combinatorics 60J10 (Primary), 60C05 (Secondary) We study $p$-colored top-$m$-to-random on the wreath product $G_{n,p}=C_p\wr S_n$, with $m$ fixed. Using the Nakano-Sadahiro-Sakurai basis elements $B_m$, we obtain exact nested-set occupancy mixtures and reduce the likelihood ratio to the single statistic $L_p$. This yields exact formulas for separation and $L^\infty(U)$, and exact one-dimensional formulas for total variation, $L^q(U)$ ($1\le q<\infty$), $χ^2$, and relative entropy. At $k=\Bigl\lfloor \frac{n}{m}(\log n+c)\Bigr\rfloor$, the number of never-chosen labels in the associated $m$-subset occupancy model converges in law to $\mathrm{Poisson}(e^{-c})$, giving the total-variation profile $f_p(c)$, the separation profile, and the corresponding $L^q(U)$, $L^\infty(U)$, $χ^2$, and relative-entropy profiles. For $m=1$ we recover colored top-to-random; for $p=1$, the total-variation profile reduces to the Diaconis-Fill-Pitman profile. For the reversed chain, we also identify optimal strong stationary times whose tail probabilities are exactly the separation distances. |
| title | Explicit cutoff profiles for colored top-$m$-to-random shuffles |
| topic | Probability Combinatorics 60J10 (Primary), 60C05 (Secondary) |
| url | https://arxiv.org/abs/2604.09933 |