Cutoff profile of the Metropolis biased card shuffling

TL;DR

This paper proves the cutoff window and profile for the Metropolis biased card shuffling, showing it converges in a manner described by the GOE Tracy-Widom distribution, confirming a previous conjecture.

Contribution

It establishes the cutoff window size and profile for the Metropolis biased card shuffling, using a novel approach involving multi-species ASEP and Hecke algebra techniques.

Findings

Cutoff window is of order N^{1/3}.

Cutoff profile matches GOE Tracy-Widom distribution.

Generalizes TASEP finishing time results.

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 Labb\'e 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 Labb\'e-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).