A two-line representation of stationary measure for open TASEP

TL;DR

The paper proves a two-line Gibbs-type representation for the stationary measure of open TASEP on a segment, showing the steady-state height distribution is the marginal of a two-line ensemble built from weighted random walks. Leveraging a DEHP matrix-method recursion, the authors establish a key identity equating the standard stationary weights with a sum over an auxiliary two-line construction, enabling detailed analysis of height fluctuations and large deviations. They derive convergence to a KPZ-fixed-point-like process on an interval for all boundary parameters and obtain a unified large-deviation principle for the height, with a single rate function that covers both fan and shock regions, connecting to and clarifying previous Derrida-type results. The framework also yields explicit expressions for normalization constants and clarifies the relationship between the two-line representation and known LDP formulas, providing a cohesive view of fluctuations and rare events in open TASEP.

Abstract

We show that the stationary measure for the totally asymmetric simple exclusion process on a segment with open boundaries is given by a marginal of a two-line measure with a simple and explicit description. We use this representation to analyze asymptotic fluctuations of the height function near the triple point for a larger set of parameters than was previously studied. As a second application, we determine a single expression for the rate function in the large deviation principle for the height function in the fan and in the shock region. We then discuss how this expression relates to the expressions available in the literature.

Figures (3)

• Figure 1: Totaly Asymmetric Simple Exclusion process with boundary parameters $\alpha,\beta$.
• Figure 2: For fixed $\mathsf a, \mathsf b$, limiting particle density $\bar{\rho}:= \lim_{N\to\infty}\tfrac{1}{N}(\tau_1+\dots+\tau_N)$ varies by region of the phase diagram for the open TASEP. These are the maximal current region, marked as $MC$, the low density region, marked as $LD$, and the high density region, marked as $HD$. Hyperbola $\mathsf a \mathsf b=1$ separates the fan region from the shock region$\mathsf a \mathsf b>1$. (These regions were identified in Derrida-Domany-Mujamel-1992.) Parameters $\mathsf a_N, \mathsf b_N$ of the TASEP in Theorem \ref{['Thm2.1']} vary with system size $N$ and converge to the triple point$( \mathsf a, \mathsf b)=(1,1)$, where the regions $MC$, $LD$, and $HD$ meet.
• Figure 3: Continuous interpolation $X_N$ is a pair of piecewise-linear lines. The first component $\frac{1}{N}\widetilde{H}_N$ of $X_N$ is marked as the thick black line and the second component of $X_N$ marked in blue. Note that $\frac{1}{N}\widetilde{H}_N(1)=\frac{1}{N}\sum_{j=1}^N\tau_j\to \bar{\rho}$, see Fig. \ref{['Fig2']}, except on the coexistence line with $\mathsf a= \mathsf b>1$. (The dashed line represents the most likely trajectory of the random curve $x\mapsto \frac{1}{N}\widetilde{H}_N(x)$ for large $N$.)

Theorems & Definitions (29)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm1.1']}
• Theorem 3.1
• proof
• Definition 3.1
• Theorem 3.2
• proof