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From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval

Wlodek Bryc, Yizao Wang, Jacek Wesolowski

Abstract

Barraquand and Le~Doussal introduced a family of stationary measures for the (conjectural) KPZ fixed point on an interval with Neumann boundary conditions, and predicted that they arise as scaling limit of stationary measures of all models in the KPZ universality class on an interval. In this paper, we show that the stationary measures for KPZ fixed point on an interval arise as the scaling limits of the height increment processes for the open asymmetric simple exclusion process in the steady state, with parameters changing appropriately as the size of the system tends to infinity.

From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval

Abstract

Barraquand and Le~Doussal introduced a family of stationary measures for the (conjectural) KPZ fixed point on an interval with Neumann boundary conditions, and predicted that they arise as scaling limit of stationary measures of all models in the KPZ universality class on an interval. In this paper, we show that the stationary measures for KPZ fixed point on an interval arise as the scaling limits of the height increment processes for the open asymmetric simple exclusion process in the steady state, with parameters changing appropriately as the size of the system tends to infinity.
Paper Structure (18 sections, 16 theorems, 184 equations, 2 figures)

This paper contains 18 sections, 16 theorems, 184 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. KPZ fixed point on an interval
  3. Open ASEP with changing parameters
  4. Main result
  5. Organization of the paper
  6. Preliminaries
  7. Askey--Wilson processes
  8. Matrix ansatz for open ASEP and Askey--Wilson processes
  9. Particle-hole duality
  10. Coupling and tightness in $D[0,1]$
  11. Proof of Theorem \ref{['thm:1']} for ${\color{blue}\mathsf a}+{\color{blue} \mathsf c}=0$
  12. Proof of Theorem \ref{['thm:1']} for ${\color{blue}\mathsf a}+{\color{blue} \mathsf c}>0$
  13. Proof of Theorem \ref{['thm:2']}
  14. Asymptotics of the numerator in \ref{['eq:Laplace']}
  15. Asymptotics of the denominator in \ref{['eq:Laplace']}
  16. ...and 3 more sections

Key Result

Theorem 1.5

Under Assumption assump:0 and under the stationary distribution $\mu_n$ with $h_n$ as in eq:h, if ${\color{blue}\mathsf a},{\color{blue} \mathsf c}$ are finite we have as processes in the space $D[0,1]$ of càdlàg functions with the Skorokhod metric and the limit process has continuous trajectories. Here ${\mathbb B}$ is a standard Brownian motion, $\eta^{({\color{blue}\mathsf a},{\color{blue} \ma

Figures (2)

  • Figure 1: Transition rates of the asymmetric simple exclusion process with open boundaries, with parameters $\alpha,\beta,\gamma,\delta, q$. The black disks represent occupied sites. The white disks represent empty sites, which represent the "holes" in the discussion of the particle-hole duality.
  • Figure 2: Phase diagram for the open ASEP with maximal current (MC), low density (LD), high density (HD) regions, and with shaded fan region $AC<1$. Our parameters converge from within the shaded fan region to the triple point (1,1).

Theorems & Definitions (37)

  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 27 more