We consider ASEP on a bounded interval and on a half-line with sources and sinks. On the full line, Bertini and Giacomin in 1997 proved convergence under weakly asymmetric scaling of the height function to the solution of the KPZ equation. We prove here that under similar weakly asymmetric scaling of the sources and sinks as well, the bounded interval ASEP height function converges to the KPZ equation on the unit interval with Neumann boundary conditions on both sides (different parameter for each side), and likewise for the half-line ASEP to KPZ on a half-line. This result can be interpreted as showing that the KPZ equation arises at the triple critical point (maximal current / high density / low density) of the open ASEP.

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter β, which is the natural dynamic associated to the usual quantum sineGordon model. It is shown that when β 2 ∈ (0, 16π 3 ) the Wick renormalised equation is well-posed. In the regime β 2 ∈ (0, 4π), the Da Prato-Debussche method [DPD02, DPD03] applies, while for3 ), the solution theory is provided via the theory of regularity structures [Hai13]. We also show that this model arises naturally from a class of 2+1-dimensional equilibrium interface fluctuation models with periodic nonlinearities.The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in [LRV13].

We introduce the strict-weak polymer model, and show the KPZ universality of the free energy fluctuation of this model for a certain range of parameters. Our proof relies on the observation that the discrete time geometric q-TASEP model, studied earlier by Borodin and Corwin, scales to this polymer model in the limit q → 1. This allows us to exploit the exact results for geometric q-TASEP to derive a Fredholm determinant formula for the strictweak polymer, and in turn perform rigorous asymptotic analysis to show KPZ scaling and GUE Tracy-Widom limit for the free energy fluctuations. We also derive moments formulae for the polymer partition function directly by Bethe ansatz, and identify the limit of the free energy using a stationary version of the polymer model.