Convergence of the Two‐Dimensional Dynamic Ising‐Kac Model to

Abstract

The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighborhood of radius 1 for 1 around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus Z 2 =.2N C 1/Z 2 for a system size N 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 nonlinear stochastic partial differential equation.This equation is the dynamic version of theˆ4 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 renormalization has to be performed in order to define the nonlinear term. Formally, this renormalization corresponds to adding an infinite mass term to the equation. We show that this need for renormalization for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value.

The Dynamic $${\Phi^4_3}$$ Φ 3 4 Model Comes Down from Infinity

“…[15,16,21] and the references therein). We showed in [31] that the 4 2 model can be obtained as the scaling limit of Ising-Kac models near criticality, as anticipated in [12]. Related results were obtained for the KPZ equation, first in [2] via a Cole-Hopf transformation, and then, following [22], in a series of works including [11,17,19,20,27,29].…”

Stochastic Allen–Cahn equation with mobility

“…Hence, the bound (4.8) combined with Calderon-Zygmund inequality readily implies, for any p ≥ 1, 27) where C > 0 depends only on ū 0 H 1 , T and p but is independent of n, ℓ, and η.…”

“…In the latter case, the potential W is not arbitrary but the quartic potential. Indeed, as shown in [4,12] for d = 1 and in [27] for d = 2, with these choices (1.1) describes the asymptotic of the fluctuations at the critical point for a Glauber dynamics with local mean field interaction. On the other hand, if we regard (1.1) as a phenomenological model for phase segregation and interface dynamics, the choice of a noise with nonzero spatial correlation length, i.e., a smooth j, is not unsound since we are going to look at the order parameter on larger space scales.…”

Tightness of the Ising–Kac Model on the Two-Dimensional Torus

“…As in [MW17a], we will let the inverse temperature β converge in a precise way as γ → 0 to β c = 1 the critical value of the mean-field system. The purpose of this paper is to prove the tightness of the magnetisation fluctuation field…”