[Paper Review] Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits

Hao Shen, Hendrik Weber|arXiv (Cornell University)|Aug 23, 2016

Markov Chains and Monte Carlo Methods24 references4 citations

TL;DR

This paper establishes the convergence of the Glauber dynamics of the 2D Kac-Blume-Capel model to singular stochastic PDEs: the dynamical $Φ^4$ equation near a curve in the $(β,\theta)$ parameter plane and the dynamical $Φ^6$ equation at a specific point on that curve. The proof uses a discrete version of the Da Prato-Debussche method with an additional coupling argument to control the linearized dynamics, confirming the emergence of universal scaling limits in statistical mechanics.

ABSTRACT

We study the Glauber dynamics of a two dimensional Blume-Capel model (or dilute Ising model) with Kac potential parametrized by $(β,θ)$ - the "inverse temperature" and the "chemical potential". We prove that the locally averaged spin field rescales to the solution of the dynamical $Φ^4$ equation near a curve in the $(β,θ)$ plane and to the solution of the dynamical $Φ^6$ equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato-Debussche method as in a result by Mourrat-Weber but an additional coupling argument is needed to show convergence of the linearized dynamics.

Motivation & Objective

• To establish the emergence of singular stochastic PDEs as scaling limits of microscopic spin systems in statistical mechanics.
• To identify the precise parameter regime in the $(\beta,\theta)$ plane where the dynamical $\Phi^4$ and $\Phi^6$ equations arise as universal limits.
• To extend the discrete Da Prato-Debussche method to the Kac-Blume-Capel model with a novel coupling argument for the linearized dynamics.
• To rigorously justify the universality of the $\Phi^{2n}_2$ equations in describing crossover regimes between Gaussian and non-Gaussian fixed points in two dimensions.

Proposed method

• Adapts the discrete Da Prato-Debussche method to handle the nonlinearities and singularities in the Glauber dynamics of the 2D Kac-Blume-Capel model.
• Implements a renormalization procedure to subtract divergent terms arising from the space-time white noise in the stochastic PDEs.
• Uses a coupling argument to control the convergence of the linearized dynamics, which is essential for the nonlinear analysis.
• Applies a local averaging procedure to the spin field to obtain a macroscopic field that converges to the solution of the SPDE.
• Analyzes the model near critical curves and points in the $(\beta,\theta)$ parameter space where the subcriticality condition holds.
• Relies on the subcriticality of the SPDEs, ensuring that small-scale behavior is dominated by linearized Gaussian dynamics.

Experimental results

Research questions

• RQ1 Under what conditions does the Glauber dynamics of the 2D Kac-Blume-Capel model converge to the dynamical $\Phi^4$ equation?
• RQ2 At which specific point in the $(\beta,\theta)$ plane does the dynamics converge to the dynamical $\Phi^6$ equation?
• RQ3 How can the discrete Da Prato-Debussche method be adapted to handle the Kac-Blume-Capel model with its non-Gaussian and non-mean-field interactions?
• RQ4 What additional analytical tools are required to control the convergence of the linearized dynamics in this setting?
• RQ5 How does the Kac potential's range affect the emergence of universal SPDE limits in two dimensions?

Key findings

• The locally averaged spin field converges to the solution of the dynamical $\Phi^4$ equation along a curve in the $(\beta,\theta)$ plane.
• At a specific point on this curve, the limiting equation is the dynamical $\Phi^6$ equation, indicating a higher-order phase transition regime.
• The convergence is established via a discrete implementation of the Da Prato-Debussche method, adapted to the Kac-Blume-Capel model.
• An additional coupling argument is required to control the linearized dynamics, which is not needed in simpler models.
• The results confirm that the $\Phi^{2n}_2$ equations describe crossover regimes between the Gaussian and Wilson-Fisher fixed points in two dimensions.
• The findings support the universality of these SPDEs as scaling limits of mean-field-type spin systems with long-range interactions.