2014/10/05 by Jean-Christophe Mourrat, Mourrat, Jean-Christophe, Hendrik Weber +1 · 1 citation
Mathematics · Physics and Astronomy · #37E20 #60H15 #60K35 #82C22 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1410.1179
openalex publication_date 2014/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Ising-Kac model is a variant of the ferromagnetic Ising model in which\neach spin variable interacts with all spins in a neighbourhood of radius\n\γ-1 for \γ \≪ 1 around its base point. We study the Glauber\ndynamics for this model on a discrete two-dimensional torus \ℤ2/\n(2N+1)\ℤ2, for a system size N \≫ \γ-1 and for an inverse\ntemperature close to the critical value of the mean field model. We show that\nthe suitably rescaled coarse-grained spin field converges in distribution to\nthe solution of a non-linear stochastic partial differential equation.\n This equation is the dynamic version of the \Φ42 quantum field theory,\nwhich is formally given by a reaction diffusion equation driven by an additive\nspace-time white noise. It is well-known that in two spatial dimensions, such\nequations are distribution valued and a Wick renormalisation has to be\nperformed in order to define the non-linear term. Formally, this\nrenormalisation corresponds to adding an infinite mass term to the equation. We\nshow that this need for renormalisation for the limiting equation is reflected\nin the discrete system by a shift of the critical temperature away from its\nmean field value.\n