Weak universality of some singular stochastic PDEs

Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes it according to the space-time scale in such a way that the system rescales to the SPDE in the macroscopic limit. We justify this belief for a class of continuous microscopic growth models (for KPZ) and phase co-existence models (for \Phi^4_3), allowing general microscopic nonlinear mechanisms beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussian random fields. This essentially allows one to reduce the problem of a general function to that of a polynomial.