Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits

We study the Glauber dynamics of a two dimensional Blume-Capel model (or dilute Ising model) with Kac potential parametrized by $(\beta,\theta)$ - the "inverse temperature" and the "chemical potential". We prove that the locally averaged spin field rescales to the solution of the dynamical $\Phi^4$ equation near a curve in the $(\beta,\theta)$ plane and to the solution of the dynamical $\Phi^6$ equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato-Debussche method as in a result by Mourrat-Weber but an additional coupling argument is needed to show convergence of the linearized dynamics.