Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of the open WASEP stationary measure without Liggett's condition

Zoe Himwich

TL;DR

The paper demonstrates that Liggett's condition is unnecessary for the convergence of open ASEP stationary measures to the open KPZ stationary measure under weak asymmetry scaling. By analyzing the Askey–Wilson process and its tangent limit, the authors show convergence to the continuous dual Hahn process, yielding the open KPZ stationary law in the limit, with the limit independent of certain boundary parameters. A Laplace-transform framework links finite-N AW representations to the CDH limit, and a coupling-based tightness argument establishes process-level convergence for the stationary height function. This work broadens the parameter regime for KPZ universality in open systems and supports conjectures about broader convergence beyond Liggett’s condition.

Abstract

We demonstrate that Liggett's condition can be relaxed without disrupting the convergence of open ASEP stationary measures to the open KPZ stationary measure. This is equivalent to demonstrating that, under weak asymmetry scaling and appropriate scaling of time and space, the four-parameter Askey-Wilson process converges to a two-parameter continuous dual Hahn process. We conjecture that the convergence of the open ASEP height function process to solutions to the open KPZ equation will hold for a wider range of ASEP parameters than those permitted by Liggett's condition.

Convergence of the open WASEP stationary measure without Liggett's condition

TL;DR

The paper demonstrates that Liggett's condition is unnecessary for the convergence of open ASEP stationary measures to the open KPZ stationary measure under weak asymmetry scaling. By analyzing the Askey–Wilson process and its tangent limit, the authors show convergence to the continuous dual Hahn process, yielding the open KPZ stationary law in the limit, with the limit independent of certain boundary parameters. A Laplace-transform framework links finite-N AW representations to the CDH limit, and a coupling-based tightness argument establishes process-level convergence for the stationary height function. This work broadens the parameter regime for KPZ universality in open systems and supports conjectures about broader convergence beyond Liggett’s condition.

Abstract

We demonstrate that Liggett's condition can be relaxed without disrupting the convergence of open ASEP stationary measures to the open KPZ stationary measure. This is equivalent to demonstrating that, under weak asymmetry scaling and appropriate scaling of time and space, the four-parameter Askey-Wilson process converges to a two-parameter continuous dual Hahn process. We conjecture that the convergence of the open ASEP height function process to solutions to the open KPZ equation will hold for a wider range of ASEP parameters than those permitted by Liggett's condition.
Paper Structure (21 sections, 19 theorems, 128 equations, 1 figure)

This paper contains 21 sections, 19 theorems, 128 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preface
  3. The Open ASEP
  4. Statement and Discussion of the Theorem
  5. Approach to proving \ref{['t:1']}
  6. Organization of the Paper
  7. Acknowledgements
  8. Background
  9. General Notation
  10. The Askey-Wilson Process
  11. Atoms in the Marginal Law
  12. Atoms and Time Reversal
  13. The Continuous Dual Hahn Process
  14. Notation for Asymptotics
  15. Convergence of the Laplace transform
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that the open ASEP parameters and height function satisfy the scaling parameters in d:scalenew. The scaling induces a tight sequence of laws of $\{h_{u,v,w,r}^{(N)}(\cdot)-h_{u,v,w,r}^{(N)}(0)\}_{N\in\mathbb{N}}$ on $(C[0,1],\mathcal{B}(C[0,1]))$ and for all $u+v>0$ and $w,r>0$, this sequence

Figures (1)

  • Figure 1: The open ASEP lattice with right, left, and boundary jump rates

Theorems & Definitions (33)

  • Theorem 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1: Proposition 2.3 CK21
  • Proposition 2.4: Section 5.14, page 378, (5.14.8) hoff
  • Proposition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • ...and 23 more