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 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 N1/3, 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 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 (Ann. Appl. Probab. 32 (2022) 753–763), which is the TASEP version (of our result).

Locations

• The Annals of Probability - View
• arXiv (Cornell University) - View - PDF