Cutoff for the Biased Random Transposition Shuffle
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909325243449344 |
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| author | Nestoridi, Evita Yan, Alan |
| author_facet | Nestoridi, Evita Yan, Alan |
| contents | In this paper, we study the biased random transposition shuffle, a natural generalization of the classical random transposition shuffle studied by Diaconis and Shahshahani. We diagonalize the transition matrix of the shuffle and use these eigenvalues to prove that the shuffle exhibits total variation cutoff at time $t_N = \frac{1}{2b} N \log N$ with window $N$. We also prove that the limiting distribution of the number of fixed cards near the cutoff time is Poisson. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cutoff for the Biased Random Transposition Shuffle Nestoridi, Evita Yan, Alan Probability Combinatorics 60J10, 05E10 In this paper, we study the biased random transposition shuffle, a natural generalization of the classical random transposition shuffle studied by Diaconis and Shahshahani. We diagonalize the transition matrix of the shuffle and use these eigenvalues to prove that the shuffle exhibits total variation cutoff at time $t_N = \frac{1}{2b} N \log N$ with window $N$. We also prove that the limiting distribution of the number of fixed cards near the cutoff time is Poisson. |
| title | Cutoff for the Biased Random Transposition Shuffle |
| topic | Probability Combinatorics 60J10, 05E10 |
| url | https://arxiv.org/abs/2409.16387 |