The dynamical Ising-Kac model in 3D converges to $\Phi^4_3$

Abstract

We consider the Glauber dynamics of a ferromagnetic Ising-Kac model on a three-dimensional periodic lattice of size $(2N + 1)3$, in which the flipping rate of each spin depends on an average field in a large neighborhood of radius $\frac1\gamma << N$. We study the random fluctuations of a suitably rescaled coarse-grained spin field as $N \to \infty$ and $\gamma \to 0$; we show that near the mean-field value of the critical temperature, the process converges in distribution to the solution of the dynamical $\Phi^4_3$ model on a torus. Our result settles a conjectured from Giacomin, Lebowitz and Presutti (DOI:10.1090/SURV/064/03). The dynamical $\Phi^4_3$ model is given by a non-linear stochastic partial differential equation (SPDE) which is driven by an additive space-time white noise and which requires renormalisation of the non-linearity. A rigorous notion of solution for this SPDE and its renormalisation is provided by the framework of regularity structures arXiv:1303.5113. As in the two-dimensional case arXiv:1410.1179, the renormalisation corresponds to a small shift of the inverse temperature of the discrete system away from its mean-field value.