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 (β,θ) – the “inverse temperature” and the “chemical potential”. We prove that the locally averaged spin field rescales to the solution of the dynamical Φ⁴ equation near a curve in the (β,θ) plane and to the solution of the dynamical Φ⁶ equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato–Debussche method [13] as in [33] but an additional coupling argument is needed to show convergence of the linearized dynamics.