Shuffling via sums of Jucys--Murphy Elements
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914578681561088 |
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| author | Arfaee, Samira Nestoridi, Evita |
| author_facet | Arfaee, Samira Nestoridi, Evita |
| contents | We consider a family of card shuffles of $n$ cards in which the allowed moves involve transpositions corresponding to the Jucys--Murphy elements of the symmetric group $\{S_m\}_{m \leq n}$. We determine the eigenvalues of the corresponding $n! \times n!$ transition matrices of these shuffles and study the mixing times for a special case, the $k$--star transpositions shuffle, a natural interpolation between the random transpositions shuffle, introduced and studied by Diaconis and Shahshahani, and the star transpositions shuffle, introduced and studied by Diaconis. We prove that the $k$--star transpositions shuffle exhibits total variation cutoff at $\frac{2n-(k+1)}{2(n-1)}n\log n$ with a window of $\frac{2n-(k+1)}{2(n-1)}n$. Furthermore, in the regimes $k/n \rightarrow 0$ or $k/n \rightarrow 1$, this shuffle has the same limit profile as random transpositions, which has been fully determined by Teyssier. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07918 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shuffling via sums of Jucys--Murphy Elements Arfaee, Samira Nestoridi, Evita Combinatorics Probability We consider a family of card shuffles of $n$ cards in which the allowed moves involve transpositions corresponding to the Jucys--Murphy elements of the symmetric group $\{S_m\}_{m \leq n}$. We determine the eigenvalues of the corresponding $n! \times n!$ transition matrices of these shuffles and study the mixing times for a special case, the $k$--star transpositions shuffle, a natural interpolation between the random transpositions shuffle, introduced and studied by Diaconis and Shahshahani, and the star transpositions shuffle, introduced and studied by Diaconis. We prove that the $k$--star transpositions shuffle exhibits total variation cutoff at $\frac{2n-(k+1)}{2(n-1)}n\log n$ with a window of $\frac{2n-(k+1)}{2(n-1)}n$. Furthermore, in the regimes $k/n \rightarrow 0$ or $k/n \rightarrow 1$, this shuffle has the same limit profile as random transpositions, which has been fully determined by Teyssier. |
| title | Shuffling via sums of Jucys--Murphy Elements |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2504.07918 |