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

TL;DR

This paper proves the convergence of the height function of open ASEP to the open KPZ fixed point's stationary measure in the shock region, extending previous results from the fan region and employing advanced integral representations.

Contribution

It establishes the convergence in the shock region of open ASEP to the open KPZ fixed point's stationary measure, using novel integral and duality techniques.

Findings

Proved convergence of open ASEP height function to open KPZ stationary measure in shock region.

Developed a new duality formula for the stationary measures in the shock region.

Extended the analysis of asymptotic behavior beyond the fan region.

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).