KPZ equation from non-simple variations on open ASEP

Abstract

This paper has two main goals. The first is universality of the KPZ equation for fluctuations of dynamic interfaces associated to interacting particle systems in the presence of open boundary. We consider generalizations on the open-ASEP from Corwin and Shen (Commun Pure Appl Math 71(10):2065–2128, 2018), Parekh (Commun Math Phys 365:569–649, 2019. https://doi.org/10.1007/s00220-018-3258-x). but admitting non-simple interactions both at the boundary and within the bulk of the particle system. These variations on open-ASEP are not integrable models, similar to the long-variations on ASEP considered in Dembo and Tsai (Commun Math Phys 341(1):219–261, 2016), Yang (Kardar–Parisi–Zhang equation from long-range exclusion processes, 2020. arXiv:2002.05176 [math.PR]). We establish the KPZ equation with the appropriate Robin boundary conditions as scaling limits for height function fluctuations associated to these non-integrable models, providing further evidence for the aforementioned universality of the KPZ equation. We specialize to compact domains and address non-compact domains in a second paper (Yang in KPZ equation from non-simple dynamics with boundary in the non-compact regime). The procedure that we employ to establish the aforementioned theorem is the second main point of this paper. Invariant measures in the presence of boundary interactions generally lack reasonable descriptions. Thus, global analyses done through the invariant measure, including the theory of energy solutions in Goncalves and Jara (Arch Ration Mech Anal 212:597, 2014), Goncalves and Jara (Stoch Process Appl 127(12):4029–4052, 2017), Goncalves et al. (Ann Probab 43(1):286–338, 2015), is immediately obstructed. To circumvent this obstruction, we appeal to the almost entirely local nature of the analysis in Yang (2020).