Abstract. We analyze a class of non-simple exclusion processes and the corresponding growth models by generalizing Gärtner's discrete Cole-Hopf transformation. We identify the main nonlinearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the Kardar-Parisi-Zhang (kpz) equation. This is the first universality result concerning interacting particle systems in the context of kpz universality class. While this class of exclusion processes are not explicitly solvable, we obtain the exact one-point limiting distribution for the step initial condition by using the previous result of [1] and our convergence result.

We prove that under a particular weak scaling, the 4-parameter interacting particle system introduced by Corwin and Petrov [10] converges to the Kardar-Parisi-Zhang (KPZ) equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated.

We show that a generalized Asymmetric Exclusion Process called ASEP(q, j) introduced in [CGRS14] converges to the Cole-Hopf solution to the KPZ equation under weak asymmetry scaling.

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In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional stochastic differential equation (SDE) corresponding to the bulk limit of Dyson's Brownian Motion, for all β ≥ 1. Our construction applies to an explicit and general class of initial conditions, including the lattice configuration {x i } = Z and the sine process. We further show the convergence of the finite to infinitedimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (Commun Math Phys 293(2):469-497, 2010) for the solution of this SDE at β = 2.