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Limits of open ASEP stationary measures near a boundary

Zongrui Yang

TL;DR

This work establishes the semi-infinite limit of open ASEP stationary measures by exploiting Askey–Wilson signed measures to obtain a tractable generating-function framework. It provides explicit total-variation convergence rates for left sublattices in both low- and high-density phases, showing convergence to a product Bernoulli measure with density 1/(1+C) in LD and to a high-density limit λ defined via AW signed measures in HD (with AC=1 yielding a product Bernoulli). The approach hinges on carefully bounding generating functions through detailed control of AW atom masses and leveraging particle–hole duality and time-reversal symmetries. The results quantify near-boundary behavior and contribute to the understanding of phase-dependent limiting structures in open ASEP, with implications for KPZ universality and nonequilibrium steady states.

Abstract

Consider the stationary measure of open asymmetric simple exclusion process (ASEP) on the lattice $\{1,\dots,n\}$. Taking $n$ to infinity while fixing the jump rates, this measure converges to a measure on the semi-infinite lattice. In the high and low density phases, we characterize the limiting measure and provide bounds on the convergence rates in total variation distance. Our approach involves bounding the total variation distance using generating functions, which are further estimated through a subtle analysis of the atom masses of Askey-Wilson signed measures.

Limits of open ASEP stationary measures near a boundary

TL;DR

This work establishes the semi-infinite limit of open ASEP stationary measures by exploiting Askey–Wilson signed measures to obtain a tractable generating-function framework. It provides explicit total-variation convergence rates for left sublattices in both low- and high-density phases, showing convergence to a product Bernoulli measure with density 1/(1+C) in LD and to a high-density limit λ defined via AW signed measures in HD (with AC=1 yielding a product Bernoulli). The approach hinges on carefully bounding generating functions through detailed control of AW atom masses and leveraging particle–hole duality and time-reversal symmetries. The results quantify near-boundary behavior and contribute to the understanding of phase-dependent limiting structures in open ASEP, with implications for KPZ universality and nonequilibrium steady states.

Abstract

Consider the stationary measure of open asymmetric simple exclusion process (ASEP) on the lattice . Taking to infinity while fixing the jump rates, this measure converges to a measure on the semi-infinite lattice. In the high and low density phases, we characterize the limiting measure and provide bounds on the convergence rates in total variation distance. Our approach involves bounding the total variation distance using generating functions, which are further estimated through a subtle analysis of the atom masses of Askey-Wilson signed measures.
Paper Structure (12 sections, 22 theorems, 119 equations, 2 figures)

This paper contains 12 sections, 22 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preface
  3. Model and results
  4. Comparision with a related result
  5. Outline of the paper
  6. Proofs of the main theorems
  7. Background
  8. Proof of Theorem \ref{['thm:LD']} and Theorem \ref{['thm:HD']}
  9. Proof of Proposition \ref{['prop:product Bernoulli']}
  10. Bounding the total variation distance by generating functions
  11. Total variation bounds of Askey--Wilson signed measures
  12. Time-reversal of Askey--Wilson signed measures

Key Result

Theorem 1.2

In the low density phase $C>A$, $C>1$, as $n\rightarrow\infty$, the open ASEP stationary measure $\mu_n$ weakly converges to the product Bernoulli measure on the semi-infinite lattice $\mathbb{Z}_+$ with density $\frac{1}{1+C}$. We furthermore assume that $A/C\notin\{q^l:l\in\mathbb{Z}_+\}$ if $A\ge where we use $\mathop{\mathrm{Ber}}\nolimits_k(\rho)$ to denote the product Bernoulli measure on th

Figures (2)

  • Figure 1: Jump rates in the open ASEP.
  • Figure 2: Phase diagrams for the open ASEP stationary measures. LD, HD, MC respectively stand for the low density, high density and maximal current phases.

Theorems & Definitions (55)

  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 1.8
  • Theorem 1.9: Theorem 1.4 in nestoridi2023approximating
  • Definition 2.1: Definition 2.1 and Definition 2.2 in wang2023askey
  • Remark 2.2
  • ...and 45 more