Abstract
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 stationarity was believed to exhibit a total-variation cutoff, and that was proved a few years ago by Labbé and Lacoin (Ann. Probab. 47 (2019) 1541–1586). In this paper, we prove that (for N cards) the cutoff window is in the order of , and the cutoff profile is given by the Tracy–Widom GOE distribution function. This confirms a conjecture by Bufetov and Nejjar (Probab. Theory Related Fields 83 (2022) 229–253). Our approach is different from (Ann. Probab. 47 (2019) 1541–1586), by comparing the card shuffling with the multispecies ASEP on , 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 (Ann. Appl. Probab. 32 (2022) 753–763), which is the TASEP version (of our result).
Funding Statement
The research of the author is supported by the Miller Institute for Basic Research in Science at the University of California, Berkeley, and NSF Grant DMS-2246664.
Acknowledgments
The author would like to thank Jimmy He and Amol Aggrawal for several valuable conversations, and Hubert Lacoin for his comments on an earlier version of this paper. The author would also like to thank the anonymous referees for carefully reading this manuscript and providing valuable feedback that helped improve the exposition. Part of this work was completed when the author was a Ph.D. student at Princeton University, Department of Mathematics.
Citation
Lingfu Zhang. "Cutoff profile of the Metropolis biased card shuffling." Ann. Probab. 52 (2) 713 - 736, March 2024. https://doi.org/10.1214/23-AOP1668
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