Mixing times for the TASEP on the circle

TL;DR

The paper determines the mixing time for the TASEP on a circle with $N$ sites and $k$ particles, showing it scales as $N^{2}\min(k,N-k)^{-1/2}$ and that the cutoff phenomenon does not occur. The authors achieve this by mapping the dynamics to a periodic last passage percolation model, analyzing flat geodesics and their coalescence, and using a novel random extension and time-shift argument to couple two evolutions. They prove both upper and lower bounds, tying the mixing behavior to KPZ-type fluctuation scales via geodesic coalescence, transversal fluctuations, and barrier constructions. The methods bridge integrable probability (periodic LPP) and interacting particle systems, providing a robust framework that extends to related models such as the TASEP with open boundaries in the maximal current phase.

Abstract

We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a circle of length $N$ with $k$ particles. We show that the mixing time is of order $N^2 \min(k,N-k)^{-1/2}$, and that the cutoff phenomenon does not occur. This confirms behavior which was separately predicted by Jara, Lacoin and Peres, and it is more broadly believed to hold for integrable models in the KPZ-universalty class. Our arguments rely on a connection to periodic last passage percolation with a detailed analysis of flat geodesics, as well as a novel random extension and time shift argument for last passage percolation.

Figures (9)

• Figure 1: The TASEP on a circle of length $7$ with $4$ particles.
• Figure 2: Visualization of the competition interfaces $(\phi_n)_{n \geq 1}$ and $(\tilde{\phi}_n)_{n \geq 1}$ within a $(9,3)$-periodic environment and initial growth interface $G_0$. The sites in $H_+$ are depicted in red, while the sites in $H_-$ are colored blue.
• Figure 3: Visualization of the path $(v,v_3,v_2,v_1,v_0)$ used in the proof of Proposition \ref{['pro:MinimalLPTiid']} and the geodesics between the endpoints of adjacent edges.
• Figure 4: Visualization of the parameters evolved in the crossing event $\mathcal{B}^{w}_{u,v}(\cdot)$ and the proof of Lemma \ref{['lem:NoCrossingLemma']}. The geodesic between $u$ and $v$ is drawn in red, while the geodesic from $v$ to $u$ via $w$ is drawn in purple.
• Figure 5: Visualization of the parameters evolved in the events $\mathcal{B}_{\textup{Con}}(\theta),\mathcal{B}_{\textup{Bar}}(\theta)$ and $\mathcal{B}_{\textup{NC}}(\theta)$ in the proof of Proposition \ref{['pro:ThreeEvents']}.

Theorems & Definitions (39)

• proof
• proof
• proof
• proof
• proof : Proof of Proposition \ref{['pro:MinimalLPTiid']}
• proof
• proof : Proof of Lemma \ref{['lem:NoCrossingLemma']}
• proof : Proof of Proposition \ref{['pro:GeodesicsPeriodic']}
• proof
• proof