Cutoff profile of the Metropolis biased card shuffling
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929634223849472 |
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| author | Zhang, Lingfu |
| author_facet | Zhang, Lingfu |
| contents | We consider the Metropolis biased card shuffling (also called the multi-species ASEP on a finite interval or the random Metropolis scan). Its convergence to stationary was believed to exhibit a total-variation cutoff, and that was proved a few years ago by Labbé and Lacoin. In this paper, we prove that (for $N$ cards) the cutoff window is in the order of $N^{1/3}$, and the cutoff profile is given by the GOE Tracy-Widom distribution function. This confirms a conjecture by Bufetov and Nejjar. Our approach is different from Labbé-Lacoin, by comparing the card shuffling with the multi-species ASEP on $\mathbb{Z}$, and using Hecke algebra and recent ASEP shift-invariance and convergence results. Our result can also be viewed as a generalization of the Oriented Swap Process finishing time convergence of Bufetov-Gorin-Romik, which is the TASEP version (of our result). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_13383 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cutoff profile of the Metropolis biased card shuffling Zhang, Lingfu Probability Discrete Mathematics Mathematical Physics Combinatorics We consider the Metropolis biased card shuffling (also called the multi-species ASEP on a finite interval or the random Metropolis scan). Its convergence to stationary was believed to exhibit a total-variation cutoff, and that was proved a few years ago by Labbé and Lacoin. In this paper, we prove that (for $N$ cards) the cutoff window is in the order of $N^{1/3}$, and the cutoff profile is given by the GOE Tracy-Widom distribution function. This confirms a conjecture by Bufetov and Nejjar. Our approach is different from Labbé-Lacoin, by comparing the card shuffling with the multi-species ASEP on $\mathbb{Z}$, and using Hecke algebra and recent ASEP shift-invariance and convergence results. Our result can also be viewed as a generalization of the Oriented Swap Process finishing time convergence of Bufetov-Gorin-Romik, which is the TASEP version (of our result). |
| title | Cutoff profile of the Metropolis biased card shuffling |
| topic | Probability Discrete Mathematics Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2208.13383 |