Stationary Distribution of open Asymmetric Simple Exclusion Processes on an Interval as a marginal of a two-layer ensemble

TL;DR

This paper addresses the open ASEP on a finite interval under Liggett's condition and provides a novel representation of its stationary distribution as the top-layer marginal of a two-layer ensemble. The authors develop a semi-explicit joint law for the two layers using a random-walk path γ and a corresponding composition σ, along with a recursion for the two-layer weight function w_σ, valid in both fan and shock regions and for q>0. They supply a detailed, case-by-case proof of the weight equations (left/right boundaries and bulk) and offer a second, Motzkin-path–based proof of the main marginal result, together with connections to existing two-layer models and special q=0 cases. The results extend previous two-layer representations beyond the fan region and provide tools for analyzing large deviations, KPZ-type limits, and stationary behavior in open ASEP through a structured two-layer framework.

Abstract

We investigate the asymmetric simple exclusion process (ASEP) on an interval with open boundaries. We provide a representation for its stationary distribution as a marginal of the top layer of a two-layer ensemble under Liggett's condition. The representation is valid in the fan region and in the shock region, extending the representation previously obtained in [Bryc-Zatitskii-2024 arXiv:2403.03275] to ASEP. We also give a recursion for the two-layer weight function.

Figures (2)

• Figure 1.1: Transition rates of the open ASEP with parameters $0\leq q<1$, $\alpha,\beta>0$, $\gamma,\delta\ge 0$.
• Figure 1.2: Left: An example of a two layer configuration as in duchi2005combinatorial for $L=10$, with labeling of the bottom row needed for \ref{['P-Duchi']}. Formula \ref{['P-Duchi']} assigns weight ${\mathcal{Q}}{\boldsymbol \tau}{\boldsymbol \xi} =\ (1+ \mathsf a)(1+ \mathsf b)^2$ to this configuration. Right: The equivalent two-layer configuration in our notation with the locations of particles ${\boldsymbol \tau}=(1,0,1,1,0,0,1,0,0,0)$ and ${\boldsymbol \xi}=(1,0,0,0,1,1,0,1,0,0)$. The random walk path ${\boldsymbol \gamma}=(0,0,0,1,2,1,0,1,0,0,0)$ is drawn as a continuous interpolation of the function $j\mapsto \gamma_j$ and gives composition ${\boldsymbol \sigma}({\boldsymbol \gamma})=(7,3,1)$. With $q=0$, formula \ref{['Q-def']} assigns the weight $Q{\boldsymbol \tau}{\boldsymbol \xi}=1$ that does not depends on $\mathsf a, \mathsf b$ to this configuration.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['thm1']}
• Remark 1.3
• Remark 1.4
• Lemma 2.1
• proof
• proof : Second proof of Theorem \ref{['thm1']}