The Ising-Kac model is a variant of the Ising model with long range interaction. We consider the Glauber dynamics on a three dimensional lattice at near critical temperature and show that, in a certain parameter regime, these dynamics approximate the parabolic $\varphi^4$ SPDE. Our result completes previous works in one and two spatial dimensions, and confirms a conjecture by Giacomin-Lebowitz-Presutti.
Technically, our analysis builds heavily on Hairer’s theory of regularity structure and more specifically the discretisation framework by Erhard and Hairer The key step is the construction and analysis of an appropriate model for the discrete particle system.
This is joint work with Paolo Grazieschi (Bath) and Konstantin Matetski (Michigan State).