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Mixing times for the open ASEP at the triple point

Patrik L. Ferrari, Dominik Schmid

TL;DR

The paper analyzes mixing times for the open ASEP on a segment at the triple point, proving subdiffusive times $t^{N}_{\text{mix}}(\varepsilon) \asymp N^{3/2+\kappa}$ for $\kappa\in[0,1/2)$ and showing poly-log refinements at $\kappa=1/2$; it does so by linking mixing to current fluctuations via a hierarchy of couplings and a censoring framework. A key technical achievement is a sharp contour-integral representation for the stationary current in the maximal current phase, together with precise asymptotics of $q$-Pochhammer expressions and Gamma-function kernels that drive the $N$-dependence. The work also establishes moderate-deviation bounds for the current and exits for second-class particles, and compares the open ASEP to the ASEP on the integers and weakly high-density regimes, yielding bounds with independent interest. The results provide rigorous subdiffusive mixing rates in a non-reversible, size-dependent boundary setting and connect them to KPZ-type fluctuations via the triple-point scaling.

Abstract

We consider mixing times for the open asymmetric simple exclusion process (ASEP) at the triple point. We show that the mixing time of the open ASEP on a segment of length $N$ for bias parameter $q$ is of order $N^{3/2+κ}$ if $1-q \asymp N^{-κ}$ for some $κ\in [0,\frac{1}{2})$, and the same result with poly-logarithmic corrections for $κ=\frac{1}{2}$. Our proof combines a fine analysis of the current of the open ASEP, moderate deviations of second class particles, the censoring inequality, and various couplings and multi-species extensions of the ASEP. Moreover, we establish a comparison between moderate deviations for the current of the open ASEP and the ASEP on the integers, as well as bounds on mixing times for the open ASEP in the weakly high density phase, which are of independent interest.

Mixing times for the open ASEP at the triple point

TL;DR

The paper analyzes mixing times for the open ASEP on a segment at the triple point, proving subdiffusive times for and showing poly-log refinements at ; it does so by linking mixing to current fluctuations via a hierarchy of couplings and a censoring framework. A key technical achievement is a sharp contour-integral representation for the stationary current in the maximal current phase, together with precise asymptotics of -Pochhammer expressions and Gamma-function kernels that drive the -dependence. The work also establishes moderate-deviation bounds for the current and exits for second-class particles, and compares the open ASEP to the ASEP on the integers and weakly high-density regimes, yielding bounds with independent interest. The results provide rigorous subdiffusive mixing rates in a non-reversible, size-dependent boundary setting and connect them to KPZ-type fluctuations via the triple-point scaling.

Abstract

We consider mixing times for the open asymmetric simple exclusion process (ASEP) at the triple point. We show that the mixing time of the open ASEP on a segment of length for bias parameter is of order if for some , and the same result with poly-logarithmic corrections for . Our proof combines a fine analysis of the current of the open ASEP, moderate deviations of second class particles, the censoring inequality, and various couplings and multi-species extensions of the ASEP. Moreover, we establish a comparison between moderate deviations for the current of the open ASEP and the ASEP on the integers, as well as bounds on mixing times for the open ASEP in the weakly high density phase, which are of independent interest.
Paper Structure (17 sections, 92 equations, 3 figures)

This paper contains 17 sections, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and results
  3. Related work
  4. Overview of the proof
  5. Outline of the paper
  6. Notation
  7. Preliminaries
  8. The asymmetric simple exclusion process
  9. The basic coupling
  10. The stationary distribution of asymmetric simple exclusion processes
  11. The censoring inequality
  12. Current fluctuations for the ASEP on the integers
  13. Current estimates in the maximal current phase
  14. Improved current estimates in the maximal current phase
  15. Identities and expansions of Gamma functions
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Visualization of the open ASEP with respect to parameters $q,\alpha,\beta,\gamma,\delta$.
  • Figure 2: Phase diagrams for the open ASEP stationary measures. The terms LD, HD, and MC, respectively, correspond to the low density, high density and maximal current phase. Moreover, we distinguish between the fan and the shock region for the open ASEP.
  • Figure 3: Plot of the function $F(\tilde{A},\tilde{C})$ used in the proof of Proposition \ref{['pro:CurrentMaxCurrent1']}.

Theorems & Definitions (12)

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  • ...and 2 more