Cutoff profile of the Metropolis biased card shuffling

TL;DR

The paper analyzes the Metropolis biased card shuffling of size $N$ and proves a total-variation cutoff with window $N^{1/3}$, whose profile is governed by the GOE Tracy-Widom distribution. The approach couples the finite-interval, multi-species ASEP to the infinite-line ASEP, leveraging height-function orderings, Hecke algebra identities, and contemporary KPZ fixed-point convergence results to transfer GOE fluctuations to the finite setting. This confirms a Bufetov–Nejjar conjecture and generalizes related results for the Oriented Swap Process/TASEP, providing a unified framework that connects combinatorial shuffling, integrable probability, and KPZ universality in mixing times. The methodology offers a pathway to extend such results to related multi-species exclusion processes and their finite-interval realizations.

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).

Figures (4)

• Figure 1: An illustration of projecting the biased card shuffling into single-species ASEPs.
• Figure 2: An illustration of the height function for an single-species ASEP configuration in a finite interval.
• Figure 3: An illustration of various processes and their height functions: the black, yellow, brown, blue functions are $h\{\xi_0^k\}$, $h\{\ddot{\zeta}^{-1+k-N}_0\}$, $h\{\dot{\zeta}_0^{-X+k-1}\}=h\{\ddot{\zeta}_0^{-X+k-1}\}$, and $h'\{\dot{\zeta}_0^k\}$, respectively. The process $\boldsymbol{\ddot{\zeta}}^{-1+k-N}$ is the 'extension' of $\boldsymbol{\xi}^k$, and $\boldsymbol{\ddot{\zeta}}^{-X+k-1}=\boldsymbol{\dot{\zeta}}^{-X+k-1}$ is the 'intermediate process'.
• Figure 4: An illustration of the skew-time reversibility, as given by e.g. (1.4) of quastel2022convergence.

Theorems & Definitions (26)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• proof : Proof of Theorem \ref{['thm:main']}: upper bound
• proof : Proof of Theorem \ref{['thm:main']}: lower bound