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From asymmetric simple exclusion processes with open boundaries to stationary measures of open KPZ fixed point: the shock region

Yizao Wang, Zongrui Yang

TL;DR

This work sustains the KPZ universality program for open systems by establishing the scaling limit of stationary open ASEP height fluctuations in the shock region AC>1. Employing an integral representation based on Askey–Wilson signed measures and a delicate duality analysis for Laplace transforms, the authors prove that the rescaled height profile converges in D([0,1]) to (1/√2)(B_x+η^{(a/√2,c/√2)}_x), where B is Brownian motion and η encodes the stationary open KPZ fixed point on an interval. The result, complementing the fan-region case, yields a complete description of second-order limit fluctuations across the full phase diagram and ties the stationary measures to a universal open KPZ fixed point structure via two representations and a Radon–Nikodym framework. The findings advance our understanding of boundary-driven KPZ universality and provide a rigorous bridge between integrable probabilistic structures (AW measures, DEHP algebra) and continuum KPZ objects on finite intervals with open boundaries.

Abstract

We continue the investigation of limit fluctuations of stationary measures of the asymmetric simple exclusion processes with open boundaries (open ASEP), complementing the recent result by Bryc et al. (2023). It was shown therein that in the fan region of the phase diagram, an appropriate scaling limit of the height function of open ASEP converges in distribution to a stochastic process introduced by Barraquand and Le Doussal (2022), known as the stationary measure of the (conjectural) open KPZ fixed point. In this paper, we establish the corresponding convergence in the shock region. Our proof is based on the integral representation of the matrix product ansatz in terms of Askey-Wilson signed measures introduced by Wang et al. (2024). The analysis of the asymptotic behavior is more delicate here than in the fan region. In particular, our proof of the duality formula for the stationary measures of the open KPZ fixed point in the shock region is different from the approach by Bryc et al. (2023) taken in the fan region: ours relies crucially on a recent result by Bryc and Zatitskii (2024).

From asymmetric simple exclusion processes with open boundaries to stationary measures of open KPZ fixed point: the shock region

TL;DR

This work sustains the KPZ universality program for open systems by establishing the scaling limit of stationary open ASEP height fluctuations in the shock region AC>1. Employing an integral representation based on Askey–Wilson signed measures and a delicate duality analysis for Laplace transforms, the authors prove that the rescaled height profile converges in D([0,1]) to (1/√2)(B_x+η^{(a/√2,c/√2)}_x), where B is Brownian motion and η encodes the stationary open KPZ fixed point on an interval. The result, complementing the fan-region case, yields a complete description of second-order limit fluctuations across the full phase diagram and ties the stationary measures to a universal open KPZ fixed point structure via two representations and a Radon–Nikodym framework. The findings advance our understanding of boundary-driven KPZ universality and provide a rigorous bridge between integrable probabilistic structures (AW measures, DEHP algebra) and continuum KPZ objects on finite intervals with open boundaries.

Abstract

We continue the investigation of limit fluctuations of stationary measures of the asymmetric simple exclusion processes with open boundaries (open ASEP), complementing the recent result by Bryc et al. (2023). It was shown therein that in the fan region of the phase diagram, an appropriate scaling limit of the height function of open ASEP converges in distribution to a stochastic process introduced by Barraquand and Le Doussal (2022), known as the stationary measure of the (conjectural) open KPZ fixed point. In this paper, we establish the corresponding convergence in the shock region. Our proof is based on the integral representation of the matrix product ansatz in terms of Askey-Wilson signed measures introduced by Wang et al. (2024). The analysis of the asymptotic behavior is more delicate here than in the fan region. In particular, our proof of the duality formula for the stationary measures of the open KPZ fixed point in the shock region is different from the approach by Bryc et al. (2023) taken in the fan region: ours relies crucially on a recent result by Bryc and Zatitskii (2024).
Paper Structure (16 sections, 19 theorems, 195 equations, 2 figures)

This paper contains 16 sections, 19 theorems, 195 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. Preface
  3. Asymmetric simple exclusion process with open boundaries
  4. Main Result
  5. Outline of the rest of the paper
  6. Askey--Wilson signed measures and open ASEP
  7. Representations of stationary measures of open KPZ fixed point
  8. A two-line representation
  9. Joint probability density function
  10. A continuity result on the Laplace transform
  11. Proof of the main theorem
  12. Asymptotics of Askey--Wilson signed measures
  13. Limit of the Laplace transform
  14. Proof of the main theorem for the remaining cases
  15. Explicit formula for the normalization constant
  16. ...and 1 more sections

Key Result

Theorem 1.3

Under Assumption assump:0, with ${\mathsf a},{\mathsf c}\in{\mathds R}$, we have as processes in $D([0,1])$, the space of càdlàg functions with Skorokhod topology billingsley99convergence. Here ${\mathds B}$ is a standard Brownian motion independent from the process $\eta^{({\mathsf a}/\sqrt 2,{\mathsf c}/\sqrt 2)}$.

Figures (2)

  • Figure 1: Jump rates in the open ASEP.
  • Figure 2: Phase diagram for the open ASEP with maximal current (MC), low density (LD), high density (HD) phases, and with shaded shock region $AC>1$. Under Assumption \ref{['assump:0']} and ${\mathsf a}+{\mathsf c}<0$, parameters converge from within the shock region to the triple point (1,1) as $n\to\infty$.

Theorems & Definitions (49)

  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • ...and 39 more