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

TL;DR

This paper proves that the fluctuation field of a spin system with Kac interaction under Glauber dynamics converges to the multivariate stochastic quantization equation , extending previous results from the Ising model to more complex models.

Contribution

It demonstrates the convergence of the fluctuation field for a Kac-interacting spin system to the  equation, confirming a conjecture for multivariate models.

Findings

Convergence of fluctuation field to  equation.

Extension from Ising model to  models.

Validation of a conjecture by Weber and Shen.

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 $\Phi^4_2$ equation. In this article we consider the case of the multivatiate stochastic 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 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 $\Phi^{2n}_2$.