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

We consider the Glauber dynamics of a ferromagnetic Ising-Kac model on a three-dimensional periodic lattice of size \((2 N + 1)^3\) ${(2N+1)}^3$ , in which the flipping rate of each spin depends on an average field in a large neighborhood of radius \(\gamma ^{-1}<\!\!< N\) ${\gamma }^{-1}<<N$ . We study the random fluctuations of a suitably rescaled coarse-grained spin field as \(N \rightarrow \infty \) $N\to \infty$ and \(\gamma \rightarrow 0\) $\gamma \to 0$ ; we show that near the mean-field value of the critical temperature, the process converges in distribution to the solution of the dynamical \(\Phi ^4_3\) ${\mathrm{\Phi }}_3^4$ model on a torus. Our result settles a conjecture from Giacomin et al. (1999). The dynamical \(\Phi ^4_3\) ${\mathrm{\Phi }}_3^4$ 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 (Hairer in Invent Math 198(2):269–504, 2014. https://doi.org/10.1007/s00222-014-0505-4). As in the two-dimensional case (Mourrat and Weber in Commun Pure Appl Math 70(4):717–812, 2017), the renormalisation corresponds to a small shift of the inverse temperature of the discrete system away from its mean-field value.