Convergence of the near critical Kac-Ising model to ${\varphi }_2^4$

The Kac-Ising model is spin model on a grid in which every spin interacts with a large number of neighbours. It is a popular model in statistical physics as it captures some aspects of the "usual" Ising model, but it is often simpler to study. We study the Glauber dynamic associated to this Kac-Ising model on a finite sub-grid of $Z^2$ near its critical temperature. We show that in a suitable scaling the locally averaged spin field is well described by the formal stochastic PDE $${\partial }_t\mathrm{\Phi }=\mathrm{\Delta }\mathrm{\Phi }-({\mathrm{\Phi }}^3-\mathrm{\infty }\mathrm{\Phi })+\xi ,$$ where $\xi$ denotes space time white noise, and the "infinite constant" appears as the limit of a renormalisation procedure.

This is joint work with J.C. Mourrat.