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It has been recently shown by H.Weber and J.C. Mourrat, for thetwo-dimensional Ising-Kac model at critical temperature, that the fluctuationfield of the magnetization, under the Glauber dynamic, converges indistribution to the solution of a non linear ill-posed SPDE: the dynamical$\Phi^4_2$ equation. In this article we consider the case of the multivatiatestochastic quantization equation $\Phi^{2n}_2$ on the two-dimensional torus,and we answer to a conjecture of H.Weber and H.Shen. We show that it ispossible to find a state space for a spin system on the two-dimensionaldiscrete torus undergoing Glauber dynamic with ferromagnetic Kac potential,such that the fluctuation field converges in distribution to $\Phi^{2n}_2$.

Convergence of Glauber dynamic on Ising-like models with Kac interaction to $\Phi^{2n}_2$