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

Abstract

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

arXiv comment50 pages