CUTOFF PROFILE OF THE METROPOLIS BIASED CARD SHUFFLING

• 成果类型: Article
• 署名作者: Zhang, Lingfu
• 署名单位: University of California System; University of California Davis
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1668
• 发表日期: 2024
• 页码: 713-736
• 关键词: simple exclusion process; random-walks; mixing times; asymptotics

摘要

We consider the Metropolis biased card shuffling (also called the multispecies 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 Labbe 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).