Convergence of the two-dimensional dynamic Ising-Kac model to $\Phi^4_2$

TL;DR

This paper proves that the two-dimensional Ising-Kac model's rescaled spin field converges to the dynamic $ ext{Phi}^4_2$ stochastic PDE, revealing the impact of renormalisation and critical temperature shifts.

Contribution

It establishes the rigorous convergence of the Ising-Kac model to the $ ext{Phi}^4_2$ equation, including the effects of renormalisation and critical temperature adjustments.

Findings

Rescaled spin field converges to $ ext{Phi}^4_2$ SPDE

Renormalisation manifests as a critical temperature shift

Provides a rigorous link between discrete model and quantum field theory

Abstract

The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighbourhood of radius $\gamma^{-1}$ for $\gamma \ll 1$ around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus $\mathbb{Z}^2/ (2N+1)\mathbb{Z}^2$, for a system size $N \gg \gamma^{-1}$ and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a non-linear stochastic partial differential equation. This equation is the dynamic version of the $\Phi^4_2$ quantum field theory, which is formally given by a reaction diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions, such equations are distribution valued and a Wick renormalisation has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation. We show that this need for renormalisation for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value.