Cutoff profile of the Metropolis biased card shuffling

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 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).

arXiv comment23 pages, 4 figures

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